Chapter 1: Light Notes - Horizon Learning

What is Reflection of Light?

The phenomenon, known as “reflection” or “reflection of light,” occurs when a light beam strikes any polished, smooth, or bright object and bounces back to our eyes. This phenomenon gives us a basis on which to view the environment. Light propagates in a straight line before, after, and during reflection. For example, the twinkling of stars or the light that a mirror reflects.

Here is the list of important terminologies used in Reflection of Light as shown in the figure above:

Normal: A perpendicular drawn to the reflecting surface at point M (known as the point of incidence) is called the normal to the reflecting surface.
Incident Ray: A ray of light that falls on the reflecting surface from a light source or an object is called the incident ray.
Reflected Ray: A ray of light that arises from the reflecting surface after reflection from it is called a reflected ray.
Angle of Incidence: The angle between the incident ray and normal to the point of incidence on the reflecting surface is known as the incident angle or Bugle of incidence. It is denoted by ∠i.
Angle of Reflection: The angle between the reflected ray and the normal to the point of incidence on the reflecting surface is known as the angle of reflection. It is denoted by ∠r.
Principal axis: It is defined as a line that divides the two mediums or the reflecting surface is called the principal axis.

Laws of Reflection

You must understand two necessary Laws of reflection after learning what it means. These rules can be used to calculate how the incident ray will reflect on different materials, such as a plane mirror, water, metal surfaces, etc. Here are the laws of reflection, as they apply to a plane mirror:

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The incident ray, the reflected ray, and the normal all lie on the same plane.
The angle of incidence (∠i) is always equal to the angle of reflection (∠r).

Types of Reflection of Light

Regular/Specular Reflection

A bright and sharp reflection, similar to those you see in mirrors, is referred to as a specular reflection. A uniform layer of a substance that is highly reflecting, such as powder, is applied to glass to create a mirror. This reflecting surface consistently reflects approximately all of the light that strikes it. The angles of reflection between different points don’t vary all that much. This indicates that the haze and blurring have been virtually gone.

Diffused Reflection

In general, reflective surfaces other than mirrors have a highly rough surface. This might be the result of surface dirt or traces of wear and tear, including scratches and dents. Even the substance the surface is constructed of might be important at times. All of this causes the reflection to lose both brightness and quality.

When compared between sites on such uneven surfaces, the angle of reflection is completely random. When rays strike rough surfaces, they are reflected in radically different directions despite incidents at slightly different spots on the surface. We can perceive non-shiny objects because of a reflection type known as diffused reflection.

Reflection of light from the Plane Mirror

The light beams are reflected back when they strike the flat mirror. The angle of incidence and reflection are equal, according to the laws of reflection. Behind the plane, which is visible in the mirror, is where the image is obtained. A reflection on a plane mirror is the method by which a virtual, erect mirror image.

The plane mirror always produces an erect, virtual image. The object’s size and the image’s size are both the same. The distance between the produced image and the mirror is the same as the position distance of the object. Images that are laterally inverted are obtained.

Differences between Regular and Irregular Reflection

Regular Reflection	Irregular Reflection
Regular Reflection occurs at smooth surfaces such as plane mirrors etc.	Irregular Reflection occurs on rough surfaces such as wood etc.
The reflected rays and the incident rays are parallel to each other after reflection, in this case.	However, in irregular reflection, both incident and reflected rays are not parallel to each other, after reflection.
An image is formed and visible after regular reflection.	But in this case, a distorted image is formed which is not clearly observed.

FAQs on Reflection of Light

Question 1: What do mean by the reflection of light?

Answer:

Reflection or Reflection of light is the process of bouncing back of light rays when it strikes the smooth and shiny reflecting surface.

Question 2: What are the two laws of reflection?

Answer:

The two laws of reflection can be stated as:

The incident ray, the reflected ray, and the normal all lie on the same plane.

The angle of incidence (∠i) is always equal to the angle of reflection (∠r).

Question 3: Name the type of reflection that happens, when a clear and sharp reflection occurs.

Answer:

Specular reflection.

Question 4: What type of image is formed by the plane mirror?

Answer:

A virtual and erect image is formed by the Plane mirror.

Question 5: What are the two types of images?

Answer:

The two types of images formed by the reflection of light are Real and Virtual images.

What is Refraction of Light?

The bending of a light wave when it passes from one medium to another due to the change in the speed of the light traveling the two different media is called the Refraction of light.

This phenomenon also occurs with sound, water, and other waves. Because of this bending of waves that are responsible for the refraction of light, we have lenses, magnifying glasses, prisms, and rainbows. Due to this phenomenon, our eyes would not be able to focus, without the refraction of light.

As shown in the above figure, light travels from Medium 1 to Medium 2. Please note that these mediums can be different materials or substances with different densities. So when an incident ray from medium 1 travels to another medium 2, the refracted ray bends either towards the normal or away from the normal (depending upon the densities of the mediums).

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Here are the definitions of important terms used to study Refraction:

Normal – The point of the surface at which an optical phenomenon occurs is called the normal. In simple words, it is termed the point of incidence. It is shown by a dotted line drawn perpendicular to the surface of the refracting material, in a ray diagram.
Incident Ray – The light rays that strike the refracting surface, at the separation of two media are called the Incident Ray.
Refracted Ray – The light rays that bend after passing into another medium are called the Refracted Ray.
Angle of Incidence – This is the angle between the incident ray and the normal. It is represented by ∠i and it is also called an Incident angle.
Angle of Refraction – This is the angle between refracted ray and the normal. It is represented by ∠r and it is also called a Refracted angle.

Laws of Refraction of Light

The refraction of light traveling through different mediums follows some laws. There are two laws of refraction as stated below which at the sight of refraction, the light follows, and we see the refracted image of the object.

The reflected, incident, and the normal at the point of incidence all will tend to lie in the same plane.
Secondly, the ratio of the sine of the angle of the incidence and refraction is constant which is termed Snell’s law.

sin i / sin r = Constant (n)

where i is the angle of incidence, r is the angle of refraction, the constant value depends on the refractive indexes of the two mediums.

What is the Refractive Index?

The Refractive index also called the index of refraction enables us to know how fast light travels through the material medium.

Refractive Index is a dimensionless quantity. For a given material or medium, the refractive index is considered the ratio between the speed of light in a vacuum (c) to the speed of light in the medium (v) on which it goes. The Refractive index for a medium is represented by small n, and it is given by the following formula:

n = c / v

where

c is the speed of the light in a vacuum, and

v is the speed of light in the medium.

The given velocities of light in different media can give the refractive index by the following also where the first medium is not vacuum:

n₂₁ = v₁ / v₂

where n₂₁is the refractive index of 2 with respect to 1.

Based on the given refractive index of the material or medium, the light ray either changes its direction or bends at the junction which separates the two given media. If the light ray travels from a certain medium to another of a slightly higher refractive index, it bends towards the normal in that case when traveling from rarer to a denser medium, or else it bends away from the normal when traveling from denser to rarer medium.

Snell’s Law

Snell’s law provides the degree or extent of refraction that occurs through a relationship between the incident angle, refracted angles, and the refractive indices of a given pair of media.

According to Snell’s law, the ratio of the sine of the incident angle to the sine of the refracted angle is a constant, for any light of a given color or for any given pair of media. The constant value is called the refractive index of the second medium with respect to the first.

Snell’s Law is given by the relation,

$\dfrac{\sin i}{\sin r}=\text{Constant}=n$

or

$\dfrac{\sin i}{\sin r}=\dfrac{v_1}{v_2}=\dfrac{n_2}{n_1}$

where,

i and r are the angle of incidence and refraction,

n is the refractive index and n₁ and n₂ are the refractive indices of medium 1 and 2, and

v₁ and v₂ are the speed of light in medium 1 and 2 respectively.

Causes of Refraction of Light

As it is known that when light travels in different mediums its speed varies. e.g. light passes through the air than in a glass. Hence, it can be said that, due to the change in the speed of light in different mediums that the light rays are refracted.

To understand the causes of refraction of light in much depth let’s understand What are rarer and denser mediums? and Types of Refractions as:

What are Rarer and Denser mediums?

Rarer medium (or Optically Rarer medium) is a medium in which the speed of light is more. For example, Air is optically rarer medium as compared to glass and water.

Denser medium (or Optically Denser medium) is a medium in which the speed of light is less. For example, Glass is optically denser medium as compared to air.

Types of Refraction

The refraction of light occurs in different ways depending on the medium through which the light travels.

Refraction from denser to rarer medium – When light rays pass through rarer to a denser medium, the light rays bend towards the normal. Due to this the angle of refraction is smaller than the angle of incidence. e.g. In the case when light rays pass from air to water or from air to glass, it bends towards normal. It is because of the reason that the speed of light rays reduces while passing from air to glass or water.
Refraction from rarer to denser medium – When light rays pass from denser to rarer medium, the light rays bend away from the normal. Due to this the angle of refraction becomes more than the angle of incidence. e.g. In case when light rays pass from water to air or glass to air, light rays bend away from the normal. The speed of light rays becomes greater while passing from glass or water to air.

Characteristics of Refraction

Some of the important characteristics of Refraction are:

The frequency of light does not change when it travels from one medium to another, but the velocity and wavelength of light changes.

A ray of light bends when it travels from one optical medium to another with a variable refractive index. For a specific pair of media, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant.

The relationship between a medium’s refractive index and the speed of light in that medium is as follows:

$\dfrac{\sin i}{\sin r}=\dfrac{v_1}{v_2}=\dfrac{n_2}{n_1}$

where,

i and r are the angle of incidence and refraction,

n is the refractive index and n₁ and n₂ are the refractive indices of medium 1 and 2, and

v₁ and v₂ are the speed of light in medium 1 and 2 respectively.

Effects of Refraction of Light

When anything interrupts the light waves, it causes refraction of the light. Light also moves mostly in the form of waves, much like most other materials.

As shown below, the pencil seems deformed in the water because light cannot travel through the water as rapidly as it can through the air. The pencil has a tiny magnification effect due to the light refraction, which makes the angle appear larger than it actually is and makes the pencil appear crooked.

Examples of Refraction of Light

The stars twinkle in the night sky due to the refraction of their light.
Looming and Mirage formation, both occur due to the optical illusions caused by the refraction of light.
The formation of rainbows in the sky and VIBGYOR, when white light passes through the prism are also major examples of refraction.
A swimming pool always seems or looks much shallower than it really is because of the light that comes from the bottom of the pool bends at the surfaces due to the refraction of light.

Applications of Refraction of Light

Refraction has many wide and common applications in optics and also in technology. A few of them are given below:

A lens uses the refraction phenomenon to form an image of an object or body for various purposes, such as magnification.
Spectacles that are worn by people with defective vision use the principle of refraction.
Refraction is used in peepholes of the house doors for safety, in cameras, inside movie projectors, and also in telescopes.

Solved Examples on Refraction of Light

Example 1: What is the constant value if the angle of incidence is 22° and the angle of refraction is given to be 15°?

Solution:

As we know,

sin i / sin r = constant

Given sin i = sin 22° and sin r = sin 22°

Putting the values of angles from log table we get

sin 22° / sin 15° = 1.44

Hence, the value of constant or refractive index is 1.44.

Example 2: What is the constant value if the angle of incidence is 30° and the angle of refraction is given to be 46°?

Solution:

Since, the

sin i / sin r = constant

Given sin i= sin 30° and sin r= sin 46°

Putting the values of angles from log table we get

sin 30° / sin 46° = 1.44

Hence, the constant is 1.44.

Example 3: What is the value of the sine of the angle of incidence if the angle of refraction is given to be sin 35°? Given the value of refractive index 1.33.

Solution:

As we know,

{sin i}/{sin r} =constant

Given constant= 1.33 and sin r = sin 35° = 0.57

Putting the values of angles from log table we get

sin i / sin 35° = 1.33

sin i = 1.33 × 0.57

= 0.75

Example 4: Calculate the speed of light in diamond with respect to air. Take the absolute refractive index of glass from the table.

Solution:

As we know we can calculate refractive index by the following formula,

n = c/v

where refractive index of diamond n= 2.42, c = 3 × 10⁸ m/s

$\therefore n= \frac{3\times 10^{8}}{v_{g}}$

$\therefore v_{d}= \frac{3\times 10^{8}}{n }$

$\therefore v_{d}= \frac{3\times 10^{8}}{2.42}$

$\therefore v_{d}= 1.24 \times 10^{8}$

Hence, the velocity or speed of light in glass is v_d = 1.24 × 10⁸ m/s

FAQs based on Refraction of light

Question 1: Define the term Refraction.

Answer:

The change that occurs in the direction of a wave when light passes from one medium to the other is known as refraction of light.

Question 2: When is the refraction of light not possible?

Answer:

When the light is incident perpendicular to the boundary or surface, refraction of light is not possible.

Question 3: What is the difference between reflection and refraction in light?

Answer:

The bouncing back of light when it strikes a smooth surface is called Reflection. While the bending of a light ray when it travels from one medium to other is called refraction of light.

Question 4: Give an example of the Refraction of light.

Answer:

There are many examples of refraction of light observed in our daily life like the Twinkling of stars. The Twinkling of stars is because of the atmospheric refraction occurs by the light from the star undergoing a gradual change in the medium.

What are Mirrors?

A mirror is a reflective surface that reflects light and creates a real or imaginary image.

When an object is placed in front of a mirror, the mirror reflects the image of the same object. The incident rays are coming from the object, and the reflected rays are what produce the image. The classification of the images as real or virtual depends on where the light rays intersect.

The two types of mirrors that are most commonly used are:

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Spherical Mirrors, and
Plane Mirrors

Plane Mirrors

The most typical mirrors are flat and are referred to as “plane mirrors.” A fine layer of silver nitrate or aluminum is deposited behind a flat piece of glass to construct plane mirrors.

When a light beam strikes a plane mirror, the light ray is reflected in such a way that it makes an equal angle with the incident ray that is, the angle of reflection is equal to the angle of incidence.

Spherical Mirrors

The curved surface we see of a shining spoon can be considered a curved mirror. The most widely used type of curved mirror is the spherical mirror.

The reflecting surface of such mirrors is considered to form a part of the surface of any sphere. Those mirrors which possess reflecting surfaces which are spherical are called spherical mirrors.

Important Terms used in Spherical Mirrors

Pole: The midpoint or the center point of the spherical mirror. It is represented by capital P. All the measurements are made from it only.
Aperture: An aperture of a mirror is a point from which the reflection of light actually takes place or happens. It also gives an idea about the size of the mirror.
Principal axis: An imaginary line that passes through the optical center and from the center of curvature of a spherical mirror. All the measurements are based on this line.
Centre of Curvature: The point in the center of the mirror surface that passes through the curve of the mirror and has the same tangent and curvature at that point. It is represented by the capital letter C.
Radius of Curvature: It is considered as the linear distance between the pole and the center of curvature. It is represented by the capital letter R,
Principal Focus: Principal Focus can be called the Focal Point also. It is present on the axis of a mirror where the rays of light parallel to the principal axis converge or appear to converge or diverge after reflection.
Focus: It is any given point on the principal axis where light rays parallel to the principal axis will converge or appear to converge after getting reflected from the mirror.

Types of Spherical Mirrors

Spherical Mirrors are categorized into two types:

Concave Mirrors
Convex Mirrors

Concave Mirror

If a hollow sphere is cut into some parts and the outer surface of the cut part is painted, then it turns out to be a mirror with its inner surface as the reflecting surface. This makes a concave mirror.

A concave mirror or converging mirror is a type of mirror that is bent towards the inwards side in the middle. Moreover, by looking in this mirror, we will feel that we are looking in a cave. We tend to use the mirror equation to deal with a concave mirror.

The equation for these mirrors determines the position of the object and the accurate size of the object. The angle of incidence in the concave mirror is not the same as the angle of reflection. Moreover, the angle of reflection, in this case, depends on the area on which the light hits.

Properties of Concave Mirrors

Light after reflection converges at a point when it strikes and reflects back from the reflecting surface of the concave mirror. Hence, it is also termed a converging mirror.
When the converging mirror is placed very near to the object, a magnified and virtual image is observed.
But, if we tend to increase the distance between the object and the mirror, then the image’s size reduces, and a real image is formed.
The image formed by the concave mirror can be small or enlarged or can be either real or virtual.

Applications of Concave Mirrors

Used in shaving mirrors: Converging mirrors are most widely used in shaving because they have reflective and curved surfaces. At the time of shaving, the concave mirror forms an enlarged as well as an erect image of the face when the concave mirror is held closer to the face.
The concave mirror used in the ophthalmoscope: These mirrors are used in optical instruments as in ophthalmoscopes for treatment.
Uses of the concave mirrors in astronomical telescopes: These mirrors are also widely used in making astronomical telescopes. In an astronomical telescope, a converging mirror of a diameter of about 5 meters or more is used as the objective.
Concave mirrors used in the headlights of vehicles: Converging mirrors are widely used in the headlights of automobiles and in motor vehicles, torchlights, railway engines, etc. as reflectors. The point light source is kept at the focus of the mirror, so after reflection, the light rays travel over a huge distance as parallel light beams of high intensity.
Used in solar furnaces: Large converging mirrors are used to focus the sunlight to produce heat in the solar furnace. They are often used in solar ovens to gather a large amount of solar energy in the focus of the concave mirror for heating, cooking, melting metals, etc.

Image Formation by Concave Mirror and their ray diagrams

When the object is placed at infinity

As the parallel rays from the object converge at the principal focus, F of a concave mirror; after reflection through it. Therefore, when the object is at infinity the image will form at F.

Properties of the image formed: Point-sized image, highly diminished in size, Real and inverted image.

When the object is placed between infinity and the Centre of Curvature

When the object is placed between infinity and the center of curvature of a concave mirror then the image is formed between the center of curvature (C) and focus (F).

Properties of image: It is diminished as compared to the object and also real and inverted.

Object at Centre of Curvature (C)

Whenever we place our object at the center of curvature (C) of a concave mirror, we get a real and inverted image formed at the same position.

Properties of image: It is of the same size as the object and also real and inverted.

The object is kept between the Centre of curvature (C) and Principal Focus (F)

When we keep the object somewhere between the center of curvature and the principal focus of the concave mirror, a real image is formed placed beyond the center of curvature (C).

Object at Principal Focus (F)

When the object is placed at the principal focus (F) of a concave mirror, a highly enlarged image of the object is formed at infinity.

Properties of image: Highly enlarged image, its nature is real and inverted

The object between Principal Focus (F) and Pole (P)

When the object is placed anywhere between the principal focus and the pole of a concave mirror, we get an enlarged, virtual and erect image formed behind the mirror.

Properties of image: Enlarged, Virtual and erect.

Position of object	Position of image	Image Size	Nature of image
Within focus( Between P and F)	Behind the mirror	Enlarged	Virtual and erect
At focus	At infinity	Highly Enlarged	Real and Inverted
Between F and C	Beyond C	Enlarged	Real and Inverted
At C	At C	Equal to object	Real and Inverted
Beyond C	Between F and C	Diminished	Real and Inverted
At Infinity	At focus (F)	Highly Diminished	Real and Inverted

Convex Mirror

If the cut part of the hollow sphere is painted from the inside, then its outer surface becomes the reflecting surface. This mirror is known as a convex mirror. A spherical mirror having its reflecting surface curved outwards is known to be a convex mirror.

The back of the mirror is shaded so that reflection only takes place from the outward bulged part. The surface of the spoon which bulged outwards can be assumed to be a convex mirror.

It is also known as a diverging mirror as the light after reflecting through its surface diverges in many directions but appears to meet at some points where the virtual, erect image of diminished size is formed.

Properties of Convex Mirror

A convex or diverging mirror is also known as a diverging mirror as this mirror diverges light when they strike its reflecting surface.
Virtual, erect, and diminished images are always formed or observed with convex mirrors, irrespective of the distance between the object and the mirror.

Application of Convex Mirror

Convex mirrors used inside buildings: Large offices, stores, and hospitals use a convex mirror to let people see around the corner so that they can avoid running into each other and prevent any collision.
The convex mirrors used in vehicles: Convex mirrors are commonly used as rear-view mirrors in the case of automobiles and vehicles because they can diverge light beams and make virtual images.
Uses of the convex mirror in a magnifying glass: These mirrors are mostly used for making magnifying glasses. In industries, to construct a magnifying glass, two convex mirrors are placed back to back.
Convex mirrors used for security purposes: Diverging mirrors are also used for security purposes in many places. They are places near ATMs so that bank customers can check if someone is behind them.
Convex mirrors are also used in various other places like street light reflectors as they can spread light over bigger areas.

Image Formation by Convex Mirror and their ray diagrams

Two possibilities of the position of the object are possible in the case of a convex mirror, which is when the object is at infinity and the object is between infinity and the pole of a convex mirror.

Object at infinity

Whenever the object is kept at infinity, we observe that a point-sized image is formed at the principal focus behind the convex mirror.

Properties of image: The image formed is highly diminished in size, virtual and erect

Object is kept between infinity and the pole

Whenever the object is kept anywhere between the infinity and the pole of a convex mirror, then we get a diminished, virtual and erect image formed between the pole and focus behind the mirror.

Properties of image: The image formed is diminished in size as well as virtual and erect.

Position of Object	Position of image	Image of Size	Nature of Image
Anywhere between pole P and Infinity	Behind the mirror between P and F	Diminished	Virtual and erect
At infinity	Behind the mirror at Focus (F)	Highly Diminished	Virtual and erect

Mirror Formula

To do the sums related to the spherical mirrors, the formula used is known as the mirror formula. It is used to calculate the focal length, image distance, object distance, and also magnification or any other thing required. We usually put the formula first and then put the signs so as to do the sums to minimize any error which can be generated.

The sign conventions which are to be followed while using the mirror formula are fixed so from the above-given diagram we can easily put the signs according to the requirement to get the required result.

The formula is given below,

$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$

where

u = object distance,

v = image distance,

f = focal length of mirror

Sign Conventions for Spherical Mirrors

Usually, if the object is located on the left side of the principal axis from the mirror then the object distance is taken negatively.
While if it is located on the right side it is taken to be positive.
The sign of focal length depends on the type of mirror we are using, as for the concave mirror it is negative and for the convex mirror on the other hand is positive always.
It is to be mentioned again that we have to follow the sign conventions strictly to get the correct answer.
Heights which are above the principal axis are positive and below are negative.

Solved Examples on Concave and Convex Mirrors

Example 1: What is the image distance in the case of a concave mirror if the object distance is 4 cm? It is given that the focal length of the mirror is 2 cm.

Solution:

As we know from mirror formula,

$\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$

Where u= object distance= -4cm

v= image distance=?

f= focal length of mirror= -2cm

Putting values we get

$\begin{aligned}\dfrac{1}{v} + \dfrac{1}{-4}&= \dfrac{1}{-2}\\\dfrac{1}{v}& = \dfrac{1}{-2}-\dfrac{1}{-4}\\v&=-4\,\text{cm}\end{aligned}$

Hence the object is located 4 cm in front of the mirror.

Example 2: What is the image distance in the case of a concave mirror if the object distance is 32 cm? It is given that the focal length of the mirror is 16 cm. State the nature and the size of the image which is formed.

Solution:

As we know from mirror formula,

$\dfrac{1} {v} + \dfrac{1} {u} = \dfrac{1} {f}$

where u = object distance= -32cm

v = image distance=?

f = focal length of mirror= -16cm

Putting values we get

$\dfrac{1}{v} + \dfrac{1}{-32} = \dfrac{1}{-16}\\\dfrac{1}{v} = \dfrac{1}{-16}-\dfrac{1}{-32}\\v= -16\text{ cm}$

Hence the object is located 8 cm in front of the mirror. And the image formed is real and inverted. As it is located at the centre of curvature hence the size of the image is also same as that of object.

Example 3: What is the image distance in the case of the convex mirror if the object distance is 12 cm? It is given that the focal length of the mirror is 12 cm.

Solution:

As we know from mirror formula,

$\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$

where u = object distance = -12 cm

v= image distance=?

f= focal length of mirror= 12cm

Putting values we get

$\frac{1}{v} + \frac{1}{-12} = \frac{1}{12}\\\frac{1}{v} = \frac{1}{12}-\frac{1}{-12}$ [Tex]\dfrac{1}{v} + \dfrac{1}{-12} = \dfrac{1}{12}\\\dfrac{1}{v} = \dfrac{1}{12}-\dfrac{1}{-12}\\ v = 6\text{ cm}[/Tex]

Hence the image is located 6cm behind the mirror. The nature of the image is virtual and erect and it is between the focus and the pole behind the mirror. The size of the image is found to be diminished.

Example 4: What is the image distance in the case of a concave mirror if the object distance is 10 cm? It is given that the focal length of the mirror is 10 cm.

Solution:

As we know from mirror formula,

$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$

Where u= object distance= -10cm

v= image distance=?

f= focal length of mirror= -10cm

Putting values we get

$\dfrac{1}{v} + \dfrac{1}{-10} = \dfrac{1}{-10}\\\dfrac{1}{v} = \dfrac{1}{-10}-\dfrac{1}{-10}\\v =\infin$

Hence the image will be formed at infinity.

Example 5: What is the image distance in the case of a convex mirror if the object is at the focus of the mirror? It is given that the focal length of the mirror is 10 cm. What can you say about the nature of the image formed and also the size?

Solution:

As we know from mirror formula,

$\frac{1} {v} + \frac{1} {u} = \frac{1} {f}$

Where u= object distance= -10cm

v= image distance=?

f= focal length of mirror= +10cm

Putting values we get

$\dfrac{1}{v} + \dfrac{1}{-10} = \dfrac{1}{10}\\\dfrac{1}{v} = \dfrac{1}{10}-\dfrac{1}{-10}\\v = 5 \text{ cm}$

Hence the image is located 5 cm behind the mirror. The image formed is virtual and erect and size is diminished.

FAQs on Concave and Convex Mirrors

Question 1: Which mirror is used by the dentist?

Answer:

A concave mirror is used by the dentist. Because a concave mirror helps to see things more wider.

Question 2: What is another name for the concave mirror?

Answer:

Due to the converging nature of the concave mirror, it can also be called a converging mirror.

Question 3: What are the applications of convex mirrors? State any two.

Solution:

Following are the two application of convex mirror:

As rear view mirrors in vehicles

As magnifying glass

Question 4: What is the difference between a concave mirror and a convex mirror?

Answer:

The spherical mirrors in which the inner side is reflecting in nature are called concave mirrors. While the spherical mirrors in which the outer side is reflecting in nature are called convex mirrors.

Spherical Mirrors

A spherical mirror or a mirror that is a part of a sphere is a mirror that has the shape of a piece that is cut out of a spherical surface or material. There exist two types of spherical mirrors which are: Concave and Convex mirrors.

Spherical Mirror Formula

The spherical mirror formula is a relation that describes how object distance (u) and image distance (v) are related to the focal length (f) of a spherical mirror. The spherical mirror equation is one of the most important relations from optics in Physics.

The spherical mirror formula is given as,

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$\dfrac{1}{f}=\dfrac{1}{v}+\dfrac{1}{u}$

Here,

f is the focal length of the mirror, which is the distance from the Principal Focus and the Pole of the spherical mirror.

v is the image distance of the mirror, which is the distance from the image formed and the pole of the spherical mirror.

u is the object distance of the mirror, which is the distance between the object and the pole of the spherical mirror.

The focal length of the mirror is equal to half of the radius of curvature of the spherical mirror and is given by the relation:

$f = \dfrac{R}{2}$

where,

f is the focal length of the spherical mirror and

R is the Radius of Curvature of the spherical mirror.

Magnification of the spherical mirror, determines how smaller or bigger the image is formed after reflection from the spherical mirror. Magnification is given either by the ratio of image and object height or by the ratio of image and object distance of the mirror.

$m = \dfrac{I}{O} = \dfrac{v}{u}$

where,

I is the Height of the Image formed,

O is the Height of the Object,

v is the image distance and

u is the object distance.

History of Spherical Mirrors in Human Civilization

As early as 30,000 years ago, people used spherical mirrors to collect water in prehistoric containers or reflect items in quiet, dark water (or utensils). The first mirrors were made from polished volcanic glass, such as obsidian, and are the oldest examples of produced mirrors. Mirrors made of obsidian that date to roughly 6000 BC have been discovered in Anatolia (now Turkey). Around 3000 BC and 4000 BC, respectively, polished copper mirrors were created in Mesopotamia and ancient Egypt.

Spherical mirrors have been utilized by human civilization for a very long time, not just in those areas. Polished stone mirrors from the year 2000 BC have also been found in Central and South America. Since 2000 BC, China has been making bronze mirrors.

Basic Terminologies for Spherical Mirrors

There are some common terms that we need to know while studying spherical mirrors, and they are as follows:

Centre of Curvature: The point in the centre of the mirror surface that passes through the curve of the mirror and has the same tangent and curvature at that point. It is represented by the capital letter C.
Radius of Curvature: It is considered the linear distance between the pole and the centre of curvature. It is represented by the capital letter R, R=2f
Principal axis: An imaginary line that passes through the optical centre and from the centre of curvature of a spherical mirror. All the measurements are based on this line.
Pole: The midpoint or the centre point of the spherical mirror. It is represented by capital P. All the measurements are made from it only.
Aperture: An aperture of a mirror is a point from which the reflection of light actually takes place or happens. It also gives an idea about the size of the mirror.
Principal Focus: Principal Focus can be called the Focal Point also. It is present on the axis of a mirror where the rays of light parallel to the principal axis converge or appear to converge or diverge after reflection.
Focus: It is any given point on the principal axis where light rays parallel to the principal axis will converge or appear to converge after getting reflected from the mirror.

Structure of Spherical Mirrors

A spherical mirror can be a concave or a convex mirror depending upon the surface of the reflection. If it is bulged out then it is a convex spherical mirror whereas if it is bent inwards it is termed as a concave spherical mirror.

A typical spherical mirror is a part of a big sphere of which the cut-out has been taken. The following diagram describes different parts which are there in a spherical mirror. The definition of these parts is already given above.

Types of Spherical Mirrors

Spherical Mirrors are of two types, namely:

Concave Mirrors
Convex Mirrors

The two types of spherical mirrors are discussed in detail:

Concave Mirror

A spherical mirror of which the reflecting surface is curved inwards which means that it faces towards the centre of the sphere is known to be a concave mirror. The back of the mirrors is always shaded so that reflection can take place only from the inward bulged surface.

The surface of the spoon which is curved inwards can be approximated to a concave mirror. It is also known as the converging mirror as the ray of light after bouncing back from it appears to converge at some points where we can obtain a real, inverted, and enlarged or diminished image based on the location of the object.

Uses of Concave Mirror

Converging mirrors are most widely used in shaving because they have reflective and curved surfaces.
A concave mirror is used in the ophthalmoscope
These mirrors are also widely used in making astronomical telescopes. In an astronomical telescope, a converging mirror of a diameter of about 5 meters or more is used as the objective.
Converging mirrors are widely used in headlights of automobiles and in motor vehicles, torchlights, railway engines, etc. as reflectors.
Large converging mirrors are used to focus the sunlight to produce heat in the solar furnace.

Convex Mirror

A spherical mirror having its reflecting surface curved outwards is known to be a convex mirror. The back of the mirror is shaded so that reflection only takes place from the outward bulged part. The surface of the spoon which is bulged outwards can be assumed to be a convex mirror. It is also known as a diverging mirror as the light after reflecting through its surface diverges to many directions but appears to meet at some points where the virtual, erect image of diminished size is formed.

Uses of Convex Mirror

Convex mirrors are used inside buildings so that people can see all around the building at once.
The convex mirror is used in vehicles. Convex mirrors are commonly used as rear-view mirrors in the case of automobiles and vehicles because they can diverge light beams and make virtual images.
These mirrors are mostly used for constructing magnifying glasses. In industries, to construct a magnifying glass, two convex mirrors are placed back to back.
Diverging mirrors are also used for security purposes in many places. They are placed near ATMs to let the bank customers check whether someone is behind them or not.
Convex mirrors are also widely used in various other places for example streetlight reflectors because they can spread light over bigger areas.

Sign Conventions for Spherical Mirrors

A set of rules that are used to set signs for terms like the object distance, image distance, focal length, etc used in spherical mirrors for mathematical analysis during the image formation are called the Sign Conventions for Spherical Mirrors.

According to the sign convention for spherical mirrors:

All distances are measured or taken from the pole of the spherical mirror.
Objects are considered to be placed on the left side of the spherical mirror.
The distances measured along the direction of the incident ray are taken as positive while, the distance measured along the direction of the reflected ray or opposite is taken as negative.

Image Formation by Spherical Mirrors

Image formed by any type of mirror can be found either where the reflected light appears to diverge from or where it converges. We have two types of spherical mirrors Concave and Convex Mirrors. Let’s discuss the image formation in each type of mirror as:

Image formation by Concave Mirror At Infinity.

Properties of the Image formed by Concave Mirrors

Point-sized image, highly diminished in size, Real and inverted image.
The parallel lines which come from the very distant object at infinity after striking the reflecting surface of the concave mirror get reflected back and meet at a point, or we can say in this case converge at a point. This point is known as the principal focus of the concave mirror.

Image formation by Convex mirror - At Infinity

Properties of the Image formed by Convex Mirrors

The image formed is highly diminished in size, virtual, and erect.
The parallel lines which come from the very distant object at infinity after striking the reflecting surface of the convex mirror get reflected back and appear to meet at a point, or we can say in this case diverge from the surface and appear to meet at a point. This point is known as the principal focus of a convex mirror.

Uses of Spherical Mirrors

Concave mirrors are used torch headlights to disperse light over a larger surface, hence enhancing the field of vision.
Convex mirrors are used in car’s rearview mirror, as it gives a wider field of view, that helps the driver to see most of the traffic behind him.

Concave mirrors are used in solar cookers as reflectors to focus sunlight on a specific area and raise the box’s temperature.
Convex mirrors are used in the rear-view mirror to provide a larger view of the road and oncoming traffic.
Concave mirrors are used in telescopes, satellite dishes, and by dentists and ENT specialists to create images of the teeth, ears, skin, and other body parts that are larger than the actual.

Uses of Spherical Mirrors - Dental Mirrrors

FAQs on Spherical Mirrors

Question 1: Write Daily Life Applications of Spherical Mirrors.

Answer:

Daily Life Applications of Spherical Mirrors are,

In Rear-view Mirrors of automobiles to see behind the car, etc.

In security mirrors used in banks, ATMs, stores, etc.

Satellites uses concave mirrors to receive and enhance signals.

Concave mirrors are used in solar cookers as reflectors to focus sunlight on a specific area and raise the temperature of the box.

Question 2: How many types of spherical mirrors are there? Name them.

Answer:

There are two types of spherical mirrors, they are:

Concave Mirror

Convex Mirror

Question 3: Define the term “radius of curvature” for spherical mirrors.

Answer:

Radius of Curvature: is considered the linear distance between the pole and the centre of curvature. It is represented by the capital letter R,

And it is related to focal length length f as,

R = 2f

Question 4: How can you define the principal focus of a concave mirror?

Answer:

When a parallel beams of light rays are incident on a concave mirror they after reflection through its surface converge at a particular point on the principal axis which is known as principal focus of concave mirror.

Question 5: How can you define the principal focus of a convex mirror?

Answer:

When a parallel beams of light rays are incident on a convex mirror they after reflection through its surface diverge through its surface but appear to meet at a particular point on the principal axis which is known as principal focus of convex mirror.

Question 6: What is Pole in Spherical Mirrors?

Answer:

The midpoint or the centre point of the spherical mirror. It is represented by capital P. All the measurements are made from it only.

Question 7: Name the mirror used in Car’s Rear-view mirror.

Answer:

Convex mirrors are used in car mirrors.

Commonly Used Terms in Spherical Mirrors

Aperture

The part of a spherical mirror that is exposed to all the light rays that incident on it is called the aperture of the spherical mirror. In other words, the diameter (XY) of the aperture of the concave mirrors and convex mirror is shown in the figure, known as its aperture.

Centre of Curvature

The center of a hollow sphere of which the curved or spherical mirror forms a part is called the center of curvature. It is denoted by C (as shown in the Figure).

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Radius of curvature

The radius of a hollow sphere of which the spherical mirror forms a part is called the radius of curvature. It is denoted by R.

Pole

The center of the spherical surface is called its pole. It is denoted by P.

Principal Axis

The line joining the center of curvature (C) and pole (P) of a spherical mirror and extended on either side is called the principal axis.

Principal Focus

A point on the principal axis of a spherical mirror where the rays of light parallel to the principal axis meet or appear to meet after reflection from the spherical mirror is called the principal focus. It is denoted by F.

In the case of a concave mirror, the rays of light parallel to the principal axis after reflection actually meet the principal axis at F as shown in figure 2. So, the principal focus of a concave mirror is real.

In the case of a convex mirror, the rays of light parallel to the principal axis after reflection appear to meet or diverge from the principal axis at F as shown in figure 3. So, the principal focus of a convex mirror is virtual.

Focal Plane

A plane normal or perpendicular to the principal axis and passing through the principal focus (F) of a spherical mirror is called the focal plane of the spherical mirror.

Focal Length

The distance between the pole (P) and principal focus (f) of a spherical mirror is called the focal length of the mirror. It is denoted by f. As shown in the figure above, the focal length of the mirror is represented by PF.

Sign Convention for Spherical Mirrors

The set of guidelines to set signs for image distance, object distance, focal length, etc for mathematical calculation during an image formation is called the Sign Convention. The sign conventions in the case of the spherical mirrors are made in taking into consideration that the objects are always placed on the left side of the mirror, such that the direction of incident light is from left to right.

The sign conventions followed for any spherical mirror are given as:

All distances are measured from the pole of a spherical error.

Distances measured in the direction of incident light are taken as positive, while distances measured in a direction opposite to the direction of the incident light are taken as negative.

The upward distances perpendicular to the principal axis are taken as positive, while the downward distances perpendicular to the principal axis is taken as negative.

For convenience, the object is assumed to be placed on the left side of a mirror. Hence, the distance of an object from the pole of a spherical mirror is taken as negative.

Since the incident light always goes from left to right, all the distances measured from the pole (P) of the mirror to the right side will be considered positive (because they will be in the same directions as the incident light). On the other hand, all the distances measured from pole (P) of the mirror to the left will be negative (because they are measured against the direction of incident light)

Important Points to Remember

According to the sign convention, the distances towards the left of the mirror are negative. Since an object is always placed to the left side of a mirror, therefore, the object distance (u) is always negative.
The images formed by a concave mirror can be either behind the mirror (virtual) or in front of the mirror (real). So, the image distance (v) for a concave mirror can be either positive or negative depending on the position of the image.
If the image is formed behind a concave mirror, the image distance (v) is positive but if the image is formed in front of the mirror, then the image distance will be negative.
In a convex mirror, the image is always formed on the right-hand side (behind the mirror), so the image distance (o) for a convex mirror will be always positive.
The focus of a concave mirror is in front of the mirror on the left side, so the focal length of a concave mirror will be negative (and written with a minus sign, say, -10 cm).
On the other hand, the focus of the convex mirror is behind the mirror on the right side, so the focal length (and written with a plus sign, say +20 cm or just 20 cm), of a convex mirror is positive.
The Focal Length and radius of curvature of a concave mirror are taken negatively.
The Focal Length and radius of curvature of a convex mirror are taken positively.

Mirror Formula

The distance of the position of an object on the principal axis from the pole of a spherical mirror is known as object distance. It is denoted by u. The distance of the position of the image of an object on the principal axis from the pole of a spherical mirror is known as the image distance. It is denoted by v.

The relation between v and f of a spherical mirror is known as the mirror formula.

It is given by,

1 / Object Distance + 1 / Image Distance = 1 / Focal Length

or

1/u + 1/v = 1/f

where

u is the object distance

v is the image distance and

f is the focal length of the mirror.

Magnification (or Linear magnification)

Linear Magnification produced by a mirror is defined as the ratio of the size (or height) of the image to the size (or height) of the object. It is denoted by m. If h’ is the size (or height) of the image produced by the mirror and h is the size (or height) of the object.

Then, Linear magnification is:

Magnification = Height of Image / Height of Object

or

m = h^‘/ h

where

m is the magnification of the spherical mirror,

h’ is the Height of Image, and

h is the Height of Object.

Linear magnification has no unit.

Solved Examples on Sign Conventions for Spherical Mirrors

Example 1: A concave mirror produces two times magnified real image of an object placed 10 cm in front of it. Find the position of the image.

Solution:

Here, -10 cm (Sign convention)

m = – 2 ( Image is real).

But m =-v/u

-2=-v/u

v=-20 cm

Thus, image of the object is at 20 cm from the pole of the mirror and in front of the mirror.

Example 2: An object of 5 cm in size is placed at a distance of 20 cm from its concave minor of the focal length of 15 cm At what distance from the mirror, should a screen be placed to get the sharp image? Also, calculate the size of the image.

Solution:

Given that,

h=+5 cm

f = – 15.0 cm (Sign convention)

u = – 20 cm (Sign convention)

Determination of the position of image.

Using,

1/u + 1/v =1/f

We get ,

1/v = 1/f-1/u

v = -60cm

So the screen must be placed at a distance of 60 cm in front of the concave mirror.

Determination of size of the image and its nature.

Using,

m= h’/h =-v/u

h^‘=-(v/u)h

= -15 cm

Thus, the size of image -15cm, negative sign with h’ shows that the image is real and inverted.

Example 3: A convex mirror used in a bus has a radius of curvature of 3.5 m. If the driver of the bus locates a car 10 m behind the bus, find the position, nature, and size of the image of the car.

Solution:

Here, R = 3-5 m f = R 2 3-5 2 = 1.75m, u = – 100 m.

Determination of the position of the car.

Using, Using, 1/u + 1/v =1/f

1/v =1/f -1/u

1/v= 1/1.75 -1/(-10)

v = 1.5 m

Thus, the car appears to be at 1.5 m from the convex mirror and behind the mirror.

Determination of the size and nature of the image

Using, m= h’/h =-v/u

= -1.5/-10

= 0.15

Thus, the size of the image of the car is 0.15 times the actual size of the car.

Since m is positive, so image of the car is virtual and erect (i.e., upright).

Example 4: Determine the focal length of the concave mirror given the radius of curvature is 20 cm.

Solution:

Given that,

The radius of curvature of the mirror, R is 20 cm.

Using the formula,

R = f/2

or

f = R × 2

= 20 cm × 2

= 40 cm

Hence, the focal length of the mirror is 40 cm.

FAQs on Sign Conventions for Spherical Mirrors

Question 1: What is the function of a Convex Mirror?

Answer:

Since, the convex mirror is a diverging mirror. So the light falls on the mirror reflects from it in the outward direction and hence the image formed behind the mirror.

Question 2: Write down the Mirror Formula.

Answer:

The mirror formula is the relation between the object, image, and focal length of the spherical mirror. The mirror formula is given as,

1/u + 1/v = 1/f

where

u is the object distance

v is the image distance and

f is the focal length of the mirror.

Question 3: What is the function of a Concave Mirror?

Answer:

Since, the concave mirror is a converging mirror. So the light falls on the mirror reflects from it in the inward direction and hence the image formed in front of the mirror.

Question 4: What is the Sign Convention followed for the Image distance of the Convex Mirror?

Answer:

In a convex mirror, the image is always formed on the right-hand side (behind the mirror), so the image distance (o) for a convex mirror will be always positive.

Question 5: What is the Sign Convention followed for the Image distance of the Concave Mirror?

Answer:

The images formed by a concave mirror can be either behind the mirror (virtual) or in front of the mirror (real). So, the image distance (v) for a concave mirror can be either positive or negative depending on the position of the image. If the image is formed behind a concave mirror, the image distance (v) is positive but if the image is formed in front of the mirror, then the image distance will be negative.

Spherical Mirrors and their Types

The curved surface we see of a shining spoon can be considered as a curved mirror. The most widely used type of curved mirror is the spherical mirror. The reflecting surface of such mirrors are considered to form a part of the surface of any sphere. Those mirrors which possess a reflecting surfaces which are spherical, are called spherical mirrors.

There are two types of spherical mirrors for which the mirror formula is used.

Concave mirror or also called the converging mirror which have an inward bent surface. They have a negative value of focal length in sign convention and can form both virtual and real images based on the position of the object.

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Position of Object	Position of Image	Image Size	Natural Image
Within focus ( Between pole P and focus F)	Behind the mirror	Enlarged	Virtual and erect
At focus	At infinity	Highly Enlarge	Real and Inverted
Between F and C	Beyond C	Enlarged	Real and Inverted
At C	At C	Equal to object	Real and Inverted
Beyond C	Between F and C	Diminished	Real and Inverted
At Infinity	At focus	Highly diminished (Pointed Size)	Real and Inverted

Convex mirror or also called the diverging mirror has a bulged outward surface. They have a positive value of focal length from sign convention. They can only form virtual and erect images which can be enlarged or diminished depending on the position of the object placed.

Position of object	Position of image	Image size	Nature of Image
Anywhere between pole P and Infinity	Behind the mirror between P and F	Diminished	Virtual and erect
At infinity	Behind the mirror at Focus	Highly diminished	Virtual and erect

Sign convention chart for reference

These are to be noted very specifically while doing sums on mirror formula as well as on magnification. Remember to put the values using the sign conventions otherwise, there will be high chances of any error while solving questions related to the spherical mirrors concave or convex.

Mirror Formula

To do the sums related to the spherical mirrors, the formula used is known as mirror formula. It is used to calculate the focal length, image distance, object distance and also the magnification or any other thing required. We usually put the formula first and then put the signs so as to do the sums to minimize any error which can be generated. The sign conventions which are to be followed while using mirror formula are fixed so from the above given diagram we can easily put the signs according to the requirement to get the required result.

Usually if the object is located on the left side of principal axis from the mirror then the object distance is taken negative. While if it is located on the right side it is taken to be positive. The sign of focal length depends on the type of the mirror we are using, as for the concave mirror it is negative and for the convex mirror on the other hand is positive always. It is to be mentioned again that we have to follow the sign conventions strictly to get the correct answer.

$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$

Where u = object distance, v = image distance, f = focal length of mirror

Magnification

Magnification is termed as the increase in the image size produced by spherical mirrors concave or convex with respect to the object size. It is considered to be the ratio of the height of the image to the height of the object and is denoted as m.

The magnification which is represented by the letter small m produced by a spherical mirror can be expressed or represented as:

$\therefore m=\frac{h^{'}}{h}$

Where the letter h is the height of the image and h’ is the height of the object.

Magnification is also equal to the ratio of the image distance from the pole of the mirror to the object distance taken from the pole of the mirror.

$\therefore m=-\frac{v}{u}$

As the object lies always above the main principal axis, the height of the object is taken always as positive. But the sign for the image height may change according to the type of image formed in case of any type of mirror chosen. The height of the virtual images formed should be taken as positive while the height of the real images formed should be taken as negative.

Uses of Magnification

A precision magnifier performs the role of a very simple magnifier, but it holds multiple elements to erase the aberrations and yield a sharper image for us.
A tiny water droplet acts as a very simple magnifier that magnifies the object present behind it. The water forms small spherical droplets due to the influence of the surface tension. When the water droplet is in contact with any object, a spherical shape is distorted but capable of forming a good image of the object.

Points to remember while calculating the magnification for spherical mirrors:

The positive magnitude or value of the magnification indicates or tells that a virtual and erect image is formed.
The negative magnitude or value of the magnification indicates or tells that a real and inverted image is formed.

Sample Problems

Question 1: What are the two types of mirrors for which the mirror formula is used?

Answer:

The two types of mirrors for which mirror formula is used are concave mirror and convex mirror.

Question 2: What is the magnification produced if the image distance is 6cm and the object is located at 12cm in case of concave mirror?

Solution:

As we know the magnification can be calculated using the following formula;

$m=-\frac{v}{u}$

Given, v= -6cm and u= -12cm the signs are given using sign convention.

$\therefore m=-\frac{-6cm}{-12cm}$

$\therefore m=-\frac{1}{2}$

m = -0.5

Hence, there is a decrease by a factor of 0.5.

Question 3: What is the image distance in case of convex mirror if the object is placed at 12cm? Determine it if the height of the image if 4cm and height of the object is 2cm.

Solution:

As we know the magnification can be calculated using the following formulas;

m $=-\frac{v}{u}$ and also m $=\frac{h'}{h}$

Given, height of image h’ = 4cm, height of object{h}= 2cm and u= -12cm the signs are given using sign convention.

$\therefore m=\frac{h'}{h}$

$\therefore m=\frac{4cm}{2cm}$

m = +2

Hence, there is an increase by a factor of 2.

$\therefore m=-\frac{v}{u}$

Putting m= 2 and u=-12cm we get

$\therefore 2=-\frac{v}{-12cm}$

$\therefore v= (-2)\times (-12cm)$

v= 24cm

Hence, the image distance is 24cm.

Question 4: What is the magnification produced if the image distance is 12cm and the object is located at 6cm in case of convex mirror?

Solution:

As we know the magnification can be calculated using the following formula;

$\therefore m=-\frac{v}{u}$

Given, v= 12cm and u= -24cm the signs are given using sign convention.

$\therefore m=-\frac{12cm}{-24cm}$

$\therefore m=-\frac{1}{-2}$

m = 0.5

Hence, there is a decrease by a factor of 0.5.

Question 5: What is the increase or decrease in the magnification if the object is located at 7cm in front of a concave mirror and the image is formed at 14cm?

Solution:

As we know the magnification can be calculated using the following formula;

m= $-\frac{v}{u}$

Given, v= -14 cm and u= -7 cm the signs are given using sign convention.

$\therefore m=-\frac{-14cm}{-7cm}$

m= $-\frac{2}{1}$

m=-2

Hence, there is an increase by a factor of 0.5.

Question 6: What is the magnification if the object height is 6cm and the image height is 24cm below the principal axis?

Solution:

As we know the magnification can be calculated using the following formula;

$m=\frac{h'}{h}$

Given, height of image h’ = -24 cm, height of object{h}= 6cm the signs are given using sign convention.

$\therefore m=\frac{h'}{h}$

$\therefore m=\frac{-24cm}{6cm}$

m=-4

Hence, the magnification is (-4).

Question 7: What is the magnification if the object height is 6cm and the image height is 18cm above the principal axis?

Solution:

As we know the magnification can be calculated using the following formula;

$m=\frac{h'}{h}$

Given, height of image h’ = 18cm, height of object{h}= 6cm the signs are given using sign convention.

$\therefore m=\frac{h'}{h}$

$\therefore m=\frac{18cm}{6cm}$

m=+3

Hence the magnification is 3.

Question 8: What is the image distance in case of concave mirror if the object distance is 8 cm? It is given that the focal length of the mirror is 4 cm.

Solution:

As we know from mirror formula,

$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$

Where u= object distance= -8cm

v= image distance=?

f= focal length of mirror= -4cm

Putting values we get

$\frac{1}{v} + \frac{1}{-8} = \frac{1}{-4}$

$\frac{1}{v} = \frac{1}{-4}-\frac{1}{-8}$

$\frac{1}{v} = \frac{1}{-4}+\frac{1}{8}$

$\frac{1}{v} = \frac{-2}{8}+\frac{1}{8}$

$\frac{1}{v} = \frac{-1}{8}$

v= -8 cm

Hence, the object is located 8 cm in front of the mirror.

Question 9: What is the image distance in case of convex mirror if the object distance is 10 cm? It is given that the focal length of the mirror is 10 cm.

Solution:

As we know from mirror formula,

$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$

Where u= object distance= -10cm

v= image distance=?

f= focal length of mirror= +10cm

Putting values we get

$\frac{1}{v} + \frac{1}{-10} = \frac{1}{10}$

$\frac{1}{v} = \frac{1}{10}-\frac{1}{-10}$

$\frac{1}{v} = \frac{1}{10}+\frac{1}{10}$

$\frac{1}{v} = \frac{2}{10}$

$\frac{1}{v} = \frac{1}{5}$

v= 5 cm

Hence, the image is located 5 cm behind the mirror.

Question 10: What is the image distance in case of concave mirror if the object distance is 11 cm? It is given that the focal length of the mirror is 11 cm.

Solution:

As we know from mirror formula,

$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$

Where u = object distance= -11cm

v= image distance=?

f= focal length of mirror= -11cm

Putting values we get

$\frac{1}{v} + \frac{1}{-11} = \frac{1}{-11}$

$\frac{1}{v} = \frac{1}{-11}-\frac{1}{-11}$

$\frac{1}{v} = \frac{1}{-11}+\frac{1}{11}$

$\frac{1}{v} = \frac{1}{0}$

$\frac{1}{v} = 0$

v= infinity

Hence, the image will be formed at infinity.

Refraction of Light

It sometimes appears or observed that when the light rays are travelling obliquely from one medium to another medium, the path or direction of the propagation of light in another or second medium somehow changes. This certain phenomenon is what is known as the refraction of light. It is a very simple word that describes the change in the velocity or speed of light when it goes or travels from one medium to another medium. The refraction of light depends upon the velocity of the material medium we use and the nature of another medium from which the light comes. There are certain laws for it also which this refraction phenomenon follows. The observable change in the velocity or speed of light rays causes refraction.

Refraction through two different media or mediums is shown above. The light ray has changed its path on travelling from air to glass, and it will again change when vice versa occurs.

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Laws Of Refraction

There are given two laws of refraction as stated below which at the sight of refraction the light follows and what we see is the refracted image formed of the object.

The refracted ray, incident or what we call obliquely falling ray and the normal ray at the point of incidence all will tend to lie together in the same plane.
Secondly, we have the ratio of the sin of the angle of the incidence and refraction as a constant or has a definite value which is stated or termed as Snell’s law.

sin i / sin r = constant

where
i is the angle of incidence,
r is the angle of refraction,

Constant value depends upon the refractive indexes of the two taken mediums. It is their ratio and is dimensionless.

What is Refractive Index?

Refractive Index first of all is a dimensionless quantity. The index of refraction or in other words refractive index enables us to know how fast light travels through the material medium. The refractive index gives an idea about the speed of light while travelling in a different medium. Whenever the light that tends to travel obliquely from one medium to another changes its direction while travelling from another, the extent of change in the direction of light rays is what we say and calculate as refractive index. The ratio of the velocities or speed of light in different media gives the refractive index.

Refractive index is of two types:

Absolute Refractive Index
Relative Refractive Index

The type of refractive index depends upon the two mediums in which the light is travelling. The absolute refractive index has one material medium and one vacuum in which the speed of light is 3 × 10⁸ m/s. The relative refractive index is the relative change in speed or velocity of light upon travelling from one given medium to another.

Upon travelling from a rarer medium to denser the light rays tend to bend towards the normal and if it travels from denser to rarer then it bends away from the normal at the point of incidence.

Absolute Refractive Index

For a considered material medium, the refractive index is observed or considered to be the ratio between the speed or velocity of light in a vacuum (c) to the speed of light in the provided material medium (v) on which it falls. The refractive index or index of refraction for a medium is represented or denoted by small n, and it is given by the following formula:

n_v = c / v

where
c is velocity of the light in vacuum
v is velocity of light in the provided medium

The velocity of light in a vacuum is 3 × 10⁸ m/s. Its speed in air is also almost the same as that in a vacuum with a minimal difference. Hence, when travelling from air to the medium the speed is taken to be 3 × 10⁸ m/s only. The absolute refractive index as the name suggests gives us a rough estimate of the optical density of the given material.

The table gives an idea about the absolute refractive indices of different mediums.

Material	Refractive Index
Air	1.0003
Water	1.333
Diamond	2.417
Ice	1.31
Ethyl Alcohol	1.36

The one with a higher refractive index is optically denser than a material with a low refractive index which becomes a rarer medium. A material having a higher optical density doesn’t mean it has a high mass by volume density as they are two different quantities. For instance, kerosene has a lower density than water, but it is optically denser than water as we can see it has a higher value of the refractive index of 1.44 than water 1.33.

Relative Refractive Index

The relative refractive index refers to the refractive index of one material medium with respect to another one. The given velocities of light in different media can give the relative refractive index by the following also where the first medium is not a vacuum:

n₂₁ = v₁ / v₂

where
n₂₁ is refractive index of the speed of light in material medium 2 with respect to the velocity of light in medium 1

Similarly,

n₁₂ = v₂ / v₁

where
n₁₂ is refractive index of the speed of light in material medium 1 with respect to the velocity of light in medium 2

On travelling from a rarer to a denser medium, a light ray bends towards normal and vice versa, and on travelling from denser to rarer it bends away from normal. As can be observed and seen that the refractive index of ice is lower than that of kerosene so the light ray after travelling from ice to kerosene has bent towards the normal and so their ratio can give us the relative refractive index.

Optical Density

A medium, which has a higher refractive index with respect to vacuum is called the optically denser medium. In an optically denser medium, the speed of light is slow in comparison to an optically rarer medium.

Calculation of Refractive Index from the Speed of Light

Refractive Index can easily be calculated when the speed of light is given in two mediums. Let’s take an example, here symbol “n” denotes the refractive index.

Speed of Light (in 1st medium) = v₁

Speed of Light (in 2nd medium) = v₂

Now, the refractive index (n) of 2nd medium with respect to 1st medium is given by,

n₂₁= Speed of Light in 1st medium / Speed of light in 2nd medium

n₂₁ = v₁/v₂

Now, the refractive index (n) of 1st medium with respect to 2nd medium is given by,

n₁₂ = Speed of Light in 2nd medium / Speed of light in 1st medium

n₁₂ = v₂ / v₁

Why is High Refractive Index Important for Optical Polymers?

High refractive index allows light rays to bend more within the material, which helps to lower the profile of the lens. Thus, by increasing the refractive index, the thickness of the lens decreases which results in lowering the weight of the Optical Polymer.

What is Refractive Index Gradient?

Refractive Index Gradient is the rate of change of the refractive index with respect to the distance travelled in the optical material. The refractive index gradient is expressed as the reciprocal of a unit of distance.

Refractive Index Gradient is the rate of change of refractive index at any point with respect to distance. Refractive Index Gradient is a vector point function.

How Does The Refractive Index Vary with Wavelength?

As we know, the speed of light is the product of frequency and wavelength. The frequency of the light wave is always constant irrespective of the medium it travels, but we know that the velocity of the light changes as it changes the medium, to compensate for the change in velocity wavelength of the light wave changes based on refraction. Hence, we can say that the refractive index of a material varies with the wavelength.

Applications on Refractive Index

Various applications of the Refractive Index are discussed below in the article.

It is widely used or applied for identifying a particular substance, confirming its purity, or measuring the given concentration.
Generally and commonly it is used for measuring the concentration of the solute in an aqueous solution. For instance, in a solution of sugar, its refractive index of it can be used to determine the sugar content.
It can also be used also in the determination of the drug concentration in the pharmaceutical or pharmacy industry.
It is widely used to calculate the focusing power of various lenses, and the dispersive power of prisms.
It is generally applied for the estimation of the thermophysical properties of many hydrocarbons and petroleum mixtures.

Also, Check

Total Internal Reflection

Lens Maker’s Formula

Solved Examples on Refractive Index

Example 1: What is the value of the sin of the angle of incidence if the angle of refraction is given to be 35°? The constant is assumed to be 1.34.

Solution:

As we know,

sin i / sin r = constant

Given constant has value = 1.34 and sin r = sin 35° = 0.57

Putting the respective values of the angles from log table we get

sin i / sin 35° = 1.34

sin i = 1.34 × 0.57

sin i = 0.763

Hence, sin of angle of incidence or sin i = 0.763.

Example 2: What is the constant value if the angle of incidence is 45° and the angle of refraction is given to be 30°?

Solution:

As we know,

sin i / sin r = constant

Given, sin i = sin 45° and sin r = sin 30°

Putting the respective values of the given angles from log table we get

sin 45° / sin 30° = (1/√2) / (1/√3)
= 1.44

Hence, the value of the constant is found to be 1.44.

Example 3: Calculate the speed of light in water and also in ice. The absolute refractive index of ice is 1.31 and water is 1.53. In which medium is the speed highest?

Solution:

As we know,

n = c/v

Refractive index of water n= 1.53, c = 3 × 10⁸ m/s

n = 3 × 10⁸/ v_g

v_g = 3 × 10⁸/ n

v_g = 3 × 10⁸/ 1.53

v_g = 1.96 × 10⁸m/s

Hence, the velocity or speed of light in water is v_w = 1.96 × 10⁸ m/s.

Second case,

Refractive index of ice n = 1.31, c = 3 × 10⁸ m/s

n = 3 × 10⁸/ v_g

v_i = 3 × 10⁸/ n

v_i = 3 × 10⁸/ 1.31

v_i = 1.96 × 10⁸m/s

Hence, the velocity or speed of light in ice is v_i = 2.29 × 10⁸ m/s.

Therefore, the velocity of light is greater in case of ice than in water.

Example 4: Calculate the speed of light in benzene. The absolute refractive index of benzene is 1.50.

Solution:

As we know we can calculate the refractive index by the following formula,

n = c/v

Refractive index of benzene n= 1.5, c = 3 × 10⁸m/s

n = 3 × 10⁸/ v_b

v_b = 3 × 10⁸/ n

v_b = 3 × 10⁸/ 1.5

v_b = 2 × 10⁸m/s

Hence, the velocity or speed of light in kerosene is v_b = 2 × 10⁸ m/s

Example 5: The velocity of light in kerosene is 2.08 × 10⁸ m/s and in water is 1.96 × 10⁸ m/s. By referring to the given values calculate or find the refractive index of the kerosene with respect to the water medium.

Solution:

As we know,

n₂₁ = v₁ / v₂

n_kw = v_w / v_k

v_k = 2.08 × 10⁸ m/s

v_w = 1.96 × 10⁸ m/s

n_kw = 2.08 × 10⁸ m/s / 1.96 × 10⁸ m/s

n_kw = 0.94

Hence, the refractive index ratio of kerosene in respect to second medium water is 0.94.

FAQs on Refractive Index

Question 1: What is a Refractive Index?

Answer:

Light travelling from one medium to another medium bends accordingly. The bending of a light ray is measured by the relative refractive index of the medium. It is also defined as the ratio of the velocity of a light ray in a vacuum to the velocity of light in the medium, It is denoted by n, n = c/v.

Question 2: What is the Unit of Refractive Index?

Answer:

The refractive index is defined as the ratio of two velocities. Thus, we can say that the refractive index is a dimensionless or unitless quantity.

Question 3: What is the refractive index of water?

Answer:

The refractive index of water with respect to vacuum is 1.333.

Question 4: What is the formula to calculate the refractive index of a medium?

Answer:

The formula to calculate the refractive index of a medium is given below,

n = c / v

where
n is the refractive index of the medium
c is the velocity of light in vacuum
v is the velocity of light in the medium

Question 5: Is the speed of light faster in glass or water?

Answer:

The refractive index of water is 1.3 and the refractive index of glass is 1.5. we know that a higher refractive index lowers the speed of light. Thus, it is evident that the speed of light is faster in water than in glass.

Laws of Refraction

Based on the refraction through different surfaces, some common properties were observed in all. There are given two laws of refraction as given and stated below which at the sight of refraction the light follows and what we see is the refracted image formed of the object.

The refracted ray from the surface, incident or what we call obliquely falling ray, and the normal ray at the point of incidence all will tend to lie together in the same plane.
We have the ratio of sin of angle of the incidence and refraction is a constant or has definite value. Which is stated as Snell’s law.
$\frac{sin \ i} {sin \ r}$ =constant
Where we have i = angle of incidence, r = angle of refraction, the constant value which depends upon the refractive indexes of the two taken mediums. It is their ratio and is dimensionless.

Sample Problems

Question 1: What is the constant value if the angle of refraction is given to be 15° and the angle of incidence is 35°?

Solution:

As we know,

$\frac{sin(i)}{sin(r)}$ =constant

Given sin i = sin35° and sin r = sin15°

Putting the respective values of the given angles from log table we get

$\therefore \frac{sin\ 35^{\degree}} {sin\ 15^{\degree}}$ = 2.19

Hence, the constant for the above-given values is 2.19.

Question 2: What is the value of the sin of angle of incidence if the angle of refraction is given to be 35°? The constant is assumed to be 1.57.

Solution:

As we know,

$\frac{sin\ i}{sin\ r}$ = constant

Given constant has value = 1.57 and sin r = 35° = 0.57

Putting the respective values of the angles from log table we get

$\therefore \frac{sin\ i} {sin\ 35^{\degree}}= 1.57$

sin i = 1.57 × 0.57

{sin i} = 0.894

Hence, sin of the angle of incidence or sin i = 0.894.

Question 3:What is the constant value if the angle of incidence is 45° and the angle of refraction is given to be 60°?

Solution:

As we know,

$\frac{sin\ i}{sin\ r}$ = constant

Given sin i = sin 45° and sin r = sin 60°

Putting the respective values of the given angles from log table we get

$\therefore \frac{sin\ 45^{\degree}} {sin\ 60^{\degree}} = 0.82$

Hence, the value of the constant is found to be 0.82.

Question 4: What will happen to the emergent ray that will come from the surface of glass slab if the angle of incidence is perpendicular or along the normal ray?

Answer:

The emergent will be along the normal itself as the incident ray will not suffer any refraction as it strikes along the normal ray at the surface of the glass slab.

Question 5: Does the light ray that travels from the surface of the glass slab undergo any dispersion or deviation?

Answer:

No, the light ray that travels through the glass slab doesn’t face any dispersion or any deviation as because the surfaces of glass slab are equal and opposite to each other hence no deviation or dispersion phenomenon occur.

Question 6: By applying the lateral displacement formula find out the lateral displacement through a glass slab which has a thickness of 5cm and the angle of incidence is 45° and refraction is 30°.

Solution:

From the given formula of lateral displacement we have,

$l = t \frac{sin\ (i-r)} {cos\ r}$

Given t = 5 cm, ∠i = 45° and ∠r = 30°

Therefore, putting the values in the above equation we have,

$\therefore l = 5 cm\frac{sin (45° - 30°)} {cos30°}$

$\therefore l = 5 cm \frac{sin (15°)} {cos30°}$

Putting respective angle values from log table we get,

$\therefore l = 5 cm \frac{ 0.25 } {0.86 }$

$\therefore l = 5 cm \times 0.29$

l = 1.45 cm

Therefore, the shift or lateral displacement is 1.45cm.

Question 7: By applying the lateral displacement formula find out the lateral displacement through a glass slab which has a thickness of 10 cm and the angle of incidence is 60° and refraction is 45°.

Solution:

From the given formula of lateral displacement we have,

$l = t \frac{sin(i-r)} {cosr}$

Given t = 10 cm, ∠i = 60° and ∠r = 45°

Therefore putting the values in the above equation we have,

$\therefore l = 10 cm \frac{sin (60° - 45°)} {cos45°}$

$\therefore l = 10cm \frac{sin 15°} {cos 45°}$

Putting respective angle values from log table we get,

$\therefore l = 10cm \frac{ 0.25 } { 0.70}$

$\therefore l = 10 cm \times 0.35$

l = 3.5 cm

Therefore, the shift or lateral displacement is 3.5cm.

Question 8: By applying the lateral displacement formula find out the thickness of a given glass slab which has a shift or lateral displacement of 5 cm and the angle of incidence is 45° and refraction is 30°.

Solution:

From the formula of lateral displacement we have,

$l = t \frac{sin(i-r)} {cosr}$

Given l = 5 cm ∠i = 45° and ∠r = 30°

Therefore putting the values in the above equation we have,

$\therefore 5 cm = t \frac{sin (45{{\degree}} - 30{{\degree}})} {cos30{{\degree}}}$

$\therefore 5 cm = t \frac{sin 15{{\degree}}} {cos30{{\degree}}}$

Putting respective angle values from log table we get,

$\therefore 5 cm = t \frac{ 0.25 } {0.86 }$

$\therefore 5 cm = t \times 0.29$

$\therefore t = \frac{5cm}{0.29}$

Therefore, the thickness of given slab is 17.24cm.

Types of Lenses

Concave lens: The lens which is thicker at the end than the middle is called the concave lens. It is also called diverging lens as it spreads out the light rays that have been refracted through it. It has the ability to diverge the parallel beam of light.

A diverging lens (Concave lens)

Convex lens: The lens which is thicker at the middle than the end is called a convex lens. It is also called a converging lens as it converges the parallel beam of light into a point.

A Converging lens (Convex lens)

Terminologies related to Spherical Lens

Pole (p): It is the middle point of the spherical lens or mirror.
Centre of curvature (C): It is the centre of the sphere from which the mirror is formed.
Principal axis: It is the lines passing through the pole and the centre of curvature of the lens.
Principal focus (F): It is the point at which a narrow beam of light converges or diverges.
Focal length (f): It is the distance between the focus and the poles of the mirror.

Image formed by the Convex Lens

There are six different cases for the image formation by a convex lens, which are discussed as:

When an object is at infinity:

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When object AB (shown in the figure below) is placed at infinity that is behind the 2F₁ of the convex mirror, the image formed after the refraction will on focus F₂ which is on the opposite side of the convex lens. The size of the image is smaller than the object and the image will be real and inverted(i.e upside down and downside up).

The image formed at – Focus (F₂)
The nature of the image formed – Real and inverted
The size of the image formed – Diminished (smaller)

When an object is at infinity, the image is formed at Focus (F₂).

When an object is placed behind the Centre of Curvature (C₁):

When the object is placed behind the centre of curvature (C₁) or behind Focus (2F₁) of the convex lens, the image formed after the refraction will be between the foci of another side of the lens (i.e. F₂ and 2F₂). The size of the image is smaller than the object. The nature of the image will be real (can be seen on the screen) and inverted( upside down).

The image formed at – Between 2F₂ and F₂.
The nature of the image formed – Real and inverted
The size of the image formed – Diminished (smaller)

When an object is placed behind the Centre of Curvature (C₁), the image is formed between 2F₂ and F₂.

When the object is placed at the centre of curvature (C₁or 2F₁):

When an object is placed at the centre of curvature (C₁) or focus (2F₁) of the convex lens, the image formed after the refraction will be on the centre of curvature (C₂) or focus (2F₂) on the other side of the lens. The size of the image is the same as the size of the object. The nature of the image is real and inverted.

The image formed at – C₂ or 2F₂.
The nature of the image formed – Real and inverted
The size of the image formed – Equal to the object size.

When the object is placed at the centre of curvature (C₁or 2F₁), the image is formed at C₂ or 2F₂.

When the object is placed between 2F₁ and F₁:

When an object is placed between the centre of curvature and the focus (F₁) of the convex lens, the image formed after reflection will be behind the centre of curvature (C₂). The size of the image will be greater than the object. The nature of the image will be real and inverted.

The image formed at – Behind centre of curvature (C₂)
The nature of the image formed – Real and inverted
The size of the image formed – Enlarged

When the object is placed between 2F₁ and F₁, the image is formed behind the centre of curvature (C₂).

When the object is placed at focus (F₁):

When an object is placed at focus (F₁) of a convex lens. The image formed after reflection will be at infinity (opposite side of the lens). The size of the object will be much larger than the object. The nature of the image will be real and inverted.

The image formed at – Infinity (opposite side of the object)
The nature of the image formed – Real and inverted
The size of the image formed – Enlarged

When the object is placed at focus (F₁), the image formed is at Infinity (opposite side of the object).

When the object is placed between pole and focus (O and F₁):

When the object is placed between the focus (F₁) and the optic centre (O) of the convex lens. The image is formed at the same side of the object behind the centre of curvature (C) or focus (F₁) of the lens. The size of the image will be larger than the object. The nature of the image will be Virtual Erect.

The image formed at – At the same side of the object behind 2F₂.
The nature of the image formed – Virtual and Erect.
The size of the image formed – Enlarged

When the object is placed between pole and focus (O and F₁), the image formed is at the same side of the object behind 2F₂.

Image formed by Concave lens

There are only two different cases for the image formation by a concave lens, which are discussed as:

When the object is placed at infinity:

When an object is placed at infinity of the concave lens (shown below). The image formed after refraction will be at the focus (F₁) on the same side of the object. The size of the image will be much smaller than the object. The nature of the image will be virtual and erect.

The image formed at – Focus (F₁)
The nature of the image formed – Virtual and Erect
The size of the image formed – Highly diminished

When the object is placed at infinity, the image formed is at focus (F₁).

When the object is placed at a finite distance from the lens:

When the object is placed at any finite distance in front of the concave lens. The image formed after refraction will be between the optic centre (O) and the focus (F) of the concave lens. The size of the image will be smaller than the object.

The image formed at – Between F₁ and optical centre
The nature of the image formed – Virtual and Erect
The size of the image formed – Diminished

When the object is placed at a finite distance from the lens, the image formed is between F₁ and the optical centre.

Sample Questions

Question 1: What is the real image?

Answer:

The image formed when rays of light meet at a certain point after reflection/refraction is real image. Real images can be displayed on screen.

Question 2: What is a virtual image?

Answer:

The image formed when rays of light appear to meet at particular point is called virtual image.

Question 3: What is a ray diagram?

Answer:

The type of diagram which helps to trace the path that light takes in order for a person to view a point on the image of an object is called a ray diagram.

Question 4: What will be the focal length of a lens, if the radius is 16 cm?

Answer:

The focal length is half of the radius of lens, i.e.

f= R / 2

= 16 cm / 2

= 8 cm

Therefore, the focal length will be 8 cm.

Question 5: What will be the focal length of a lens when it is cut along the principal axis?

Answer:

There will be no change in the focal length of a lens when cut into two halves along the principal axis, because the focal length of the lens is half of the radius of curvature and radius of curvature will remain the same.

What is Power of Lens?

The ability of a lens to bend light is really what gives it its power in Ray Optics. The greater a lens’s strength, the greater its ability to refract light passing through it. Power defines the converging ability of a convex lens and the diverging ability of a concave lens. The number of light bends increases as the focal length reduces. As a result, we can assume that the strength of a lens is inversely proportional to its focal length. A short focal length, in essence, leads to high optical power. Thus,

P = 1 / f

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Here, P is the power of the lens and f is the focal length of the lens. If the focal length is given in meters (m), the power of the lens is calculated in Diopters (D), as the lens’s unit of power is diopter.

Power of Lens Formula

Mathematically formula for the power of a lens is defined as:

Power (P) = 1 / Focal Length (f)

P = 1 / f

For instance, if the focal length of a lens is 15 cm, we get 0.15 m when we translate it to meters. Take the reciprocal of 0.15 to get the power of this prism, which is 6.67. As a result, the power of this lens is 6.67 D. This assumes that the power of a lens can be calculated using the radii of curvature of two surfaces and the refractive index of the lens material.

The power of a lens is inversely proportional to its focal length. Therefore, a short focal length lens has more power, whereas a lens of long focal length has less power.

Important Points on Power of Lens:

Power of a Convex Lens (Converging Lens) is positive as its focal length is positive.

Power of a Concave Lens (Diverging Lens) is negative as its focal length is negative.

Power of a plane glass is 0.

Example 1: How does the power of a lens change if its focal length is doubled?

Solution:

Power gets halved as Power is inversely proportional to focal length.

Example 2: What is the power of a convex lens (with sign) of focal length 40cm?

Solution:

Since, Power = 1 / f

Substituting the given values as,

P = 100/40

= 2.5D

Since it’s a convex lens, so power will be positive.

Thus, the power of convex lens is +2.5D.

Example 3: Identify the type of lens, and its focal length if its power is 0.2D.

Solution:

Since the focal length, f = 1 / Power (P)

Therefore, substituting the given values in the above expression as:

f = 1 / (0.2D)

= 5 m

Since the power is positive, therefore given lens is a convex lens.

Example 4: A convex lens of a focal length of 50 cm is in contact with a concave lens of 20 cm focal length. Find the power of the combination of lenses.

Solution:

For a combination of lenses,

P = P₁ + P₂

P = 1/f₁ + 1/f₂

P = 100/50 + 100/(-20) (concave lenses have negative focal length)

= -3 D

Example 5: Find the power of a plano-convex lens, when the radius of a concave surface is 10 cm and refractive index is (n) 1.5.

Solution:

Given
R₁ = ∞, R₂ = – 10 cm = -0.1 m, n= 1.5

P = (n-1) × (1/R₁ – 1/R₂)

P = (1.5-1) × (1/∞ – 1/{-0.1})

P = 0.5 × (0 + 10)

P = 5 D

FAQs on Power of Lens

Question 1: What is the power of a lens?

Answer:

Power of a lens defines the degree of convergence and divergence of the light that strikes the lens. Thus, variation in power changes the convergence and divergence of the light that passes through the lens.

Question 2: What is the relation of focal length with the power of the lens?

Answer:

Power of the lens is reciprocal to the focal length of the lens.

Question 3: What is the SI unit for the power of a lens?

Answer:

SI unit for the power of the lens is Diopter. Its dimension is m^-1.

Question 4: Which lens has positive power?

Answer:

The power of the Converging lens or Convex lens is always positive.

Question 5: Sunglasses have curved surfaces but still do not have any power. Why?

Answer:

Sunglasses have two curved surfaces, one is convex and another is concave. Both surfaces are of equal power but of opposite signs, so both the power cancels out each other and resultant power is 0.