Full Notes

Terms Associated with Spherical Mirrors

Image Formation by Spherical Mirrors

Rules for Construction of Ray Diagrams for Spherical Mirrors

Rule 1: An incident light ray parallel to the principal axis,  passes through the principal focus or appears to pass through the principal focus after reflection.

Rule 2: An incident light ray that passes through the principal focus or appears to pass through the principal focus, travel parallel to the principal axis after reflection.

Rule 3: An incident light ray that  passes through the center of curvature or appears to pass through the center of curvature, after reflection and retraces its initial path. 

Rule 4: A ray incident obliquely to the principal axis towards  the pole, P of the curved mirror (concave mirror and convex mirror) is reflected obliquely.

Note:

The incident and reflected rays always follow the laws of reflection at the point of incidence (P) making equal angles with the principal axis.

that  passes through the center of curvature or appears to pass through the center of curvature, after reflection and retraces its initial path. 

Reflection By Concave Mirrors

Reflection by Convex Mirror

Concave Mirror

 Terms Associated With Concave Mirror

The geometric centre of a concave mirror is called its pole. 

The centre of the sphere from which the concave mirror was cut is called the centre of curvature of the concave mirror. 

The centre of curvature of the reflecting surface of a concave mirror is called the centre of curvature of the concave mirror. 

The distance from any point on the concave mirror to its center of curvature is called the radius of curvature of the concave mirror. 

An imaginary line passing through the center of curvature and the pole of the concave mirror is called the principal axis of the concave mirror. 

The area of a concave mirror that is exposed to incident light is called the aperture of the concave mirror.

The length along the principal axis from the pole to the principal focus is called the focal length of the concave mirror.

 If an object is placed close to a concave mirror such that the distance between the mirror and the object is less than its focal length, then a magnified and virtual image is formed. 

This property of the concave mirror is used in many applications such as a dentist mirror to view the inner parts of the mouth clearly and a shaving mirror.

 Concave mirrors converge the light incident on them and hence are called converging mirrors.

Image Formation by Concave Mirror

Location of an image of an object formed by a concave mirror by drawing the ray diagrams.

*We can locate the image of an object formed by a concave mirror by drawing the ray diagrams.

*The intersecting point of at least two reflections will give the position of image of the point object.

*The following rays can be used to draw the ray diagrams.

*A ray parallel to the principal axis of a concave mirror.

*A ray passing through the focus of the concave mirror

*A ray which is passing through the centre of curvature of a concave mirror

*A ray incident obliquely to the principal axis on a concave mirror.

Rules for Drawing Ray Diagrams in Spherical Concave Mirrors

A ray parallel to the principal axis of a concave mirror.

*A ray parallel to the principal axis of the concave mirror reflects through its focus.

A ray passing through the focus of the concave mirror.

*A ray passing through the focus of the concave mirror reflects parallel to the principal axis.

A ray which is passing through the centre of curvature of a concave mirror

*A ray which is passing through the centre of curvature of a concave mirror reflects back on the same path.

*A ray incident obliquely to the principal axis on a concave mirror.

A ray when incident obliquely to the principal axis on a concave mirror also reflects obliquely.

Concave Mirrors – Ray Diagrams 

Depending on the position of the object in front of the concave mirror, the position, size and the nature of the image varies.

We can represent the images formed by a Concave Mirror using Ray Diagrams.

Object at infinity

A real, inverted, highly diminished image is formed at the focus, F, in front of the concave mirror.

Object beyond C

A real, inverted, diminished image is formed between C and F, in front of the concave mirror.

Object at C

A real, inverted, same sized image is formed at C, in front of the concave mirror.

Object between C and F

A real, inverted, enlarged image is formed beyond C, in front of the concave mirror.

Object at F

A real, inverted, highly enlarged image is formed at infinity, in front of the concave mirror.

Object between F and P

A virtual, erect and enlarged image is formed behind the concave mirror.

Image Formation by a Concave Mirror

Object Location	Image Location	Nature of Image
Infinity	At F	Real, InvertedHighly DiminishedMagnification<<1
Beyond C	Beyond F and C	Real, Inverted, Diminished Magnification<1
At C	At C	Real, Inverted, Equal to size of object Magnification=1
Between C and F	Beyond C	Real, Inverted, Magnified Magnification>1
At F	Infinity	Real, Inverted, Highly Magnified Magnification>>1
Between F and P	Behind the mirror	Virtual, Erect, Magnified, Magnification>1

Uses of Concave Mirrors

• Concave mirrors are used as shaving mirrors to see a larger image of the face.
• Dentists use concave mirrors to view a magnified view of the interior parts of the mouth.
• ENT doctors use them for examining the internal parts of the ear, nose and throat.
• They are used as reflectors in the headlights of vehicles, searchlights and in torch lights to produce a strong parallel beam of light.
• Huge concave mirrors are used to focus sunlight to produce heat in solar furnaces.

Convex Mirror

A spherical mirror whose reflecting surface is curved outward is known as a convex mirror.

Terms Associated With Convex Mirror

The geometric centre of the curvature of the convex mirror is called its pole. 

The centre of curvature of the reflecting surface of a convex mirror is called the centre of curvature of the convex mirror. 

The distance from any point on the reflecting surface of a convex mirror to its centre of curvature is called radius of curvature of the convex mirror. 

An imaginary line passing through the centre of curvature and the pole of the convex mirror is called the principal axis of the convex mirror.

The reflected rays, when projected backwards, appear to meet at a point on the principal axis. This point is called the principal focus. The length along the principal axis from the pole to the principal focus is called the focal length of the concave mirror.

The area of a convex mirror that is exposed to incident light is called the aperture of the convex mirror. 

If the aperture of a convex mirror is small, then Convex mirrors, such as the rear view mirrors of cars and bikes, always form erect, virtual, and diminished images. 

The location of the object does not affect the characteristics of the image formed by a convex mirror. 

When an object approaches a convex mirror, the image formed by the mirror also approaches the mirror, but not proportionately. Because of this it is mentioned as “Objects seen in the mirror are closer than they appear” on the outside rear view mirrors of vehicles.

Image Formation by Concave Mirror

ray diagrams – 

• We can locate the image of an object formed by drawing a ray diagram. 
• The intersecting point of at least two reflections will give the position of image of the point object.

The two rays that can be used to draw the ray diagram are:

• A ray parallel to the principal axis.
• 
• A ray parallel to the principal axis reflects
• It passes through the focus in case of a concave mirror.
• A ray parallel to the principal axis. On reflection it appears to diverge from principal focus after reflection in case of a convex mirror. 

• A ray passing through the focus of the concave mirror. On reflection it becomes parallel to the principal axis due to reflection.
• A ray directed towards the focus of convex mirrors. On reflection it  becomes parallel to the principal axis due to reflection.

• A ray which is passing through the centre of curvature of a concave mirror. It reflects back on the same path.
• 
• A ray which is directed towards the centre of curvature of a convex mirror. It reflects back on the same path.

• A ray when incident obliquely to the principal axis on a concave x mirror is also reflected obliquely.
• 
• A ray when incident obliquely to the principal axis on a convex mirror is also reflected obliquely.

Reflection by Convex Mirror

Image Formed By A Convex Mirror

Irrespective of the position of the object, a virtual, erect and diminished image is formed between F and P, behind the convex mirror. 

Uses of Convex Mirrors

Convex mirrors are used as:

• rear view mirrors in automobiles and in ATM centres as it covers a wide area behind the driver.
• reflectors in street light bulbs as it diverges light rays over a wide area.
• Rear view mirrors of vehicles and the ones used.

Sign Convention for Spherical Mirrors

• Object is always considered at the left side of the mirror
• Distances measured in the direction of the incident ray are taken as positive.
• Distances measured in the direction opposite to that of the incident rays are taken as negative.
• All distances are measured from the pole of the mirror.
• Distances measured along the y-axis above the principal axis are taken as positive.
• Distances measured along the y-axis below the principal axis are taken as negative.

Table Showing Sign Convention

Mirror Formula

The relation between the focal length (f), object distance (u) and the image distance (v) is given by:

Magnification

The ratio of the height of the image in a spherical mirror, to the height of the object is called magnification (m)

Magnification,m = Height of Image, HHeight of Object, i= Image Distance, HObject Distance, i

 The distance from the principal focus to the pole of the mirror is the focal length of the mirror and is equal to half the radius of curvature, which is the distance between the centre of curvature and the pole.

For a real image object distance (u) and the image distance (v) are negative and the magnification is negative. 

If the  magnification of an image is negative it does mean that the image is real and inverted.

On the other hand for a virtual image object distance (u) is negative and image distance (v) is positive and hence the magnification is positive, i.e., the image is erect. 

If the  magnification of an image is positive it does mean that the image is virtual and erect.

• If the magnification is less than 1 The image formed is diminished in size.
• If the magnification is more than 1 The image formed is magnified in size.
• If the magnification is equal to 1 The image formed is  equal to the object in size.

Convex mirrors diverge the light incident on them and hence they are called the diverging mirrors. 

Due to this they always form diminished, virtual and erect images irrespective of the position of the object in front of them. 

Thus, the magnification caused by these mirrors is always less than one. 

The field of view for a convex mirror is greater than that for a plane mirror, the aperture being the same. 

Hence, convex mirrors are used as rear-view mirrors in vehicles. 

It is also installed behind automated teller machines as a security measure.

 The field of view for a convex mirror is greater than that for a plane mirror, the aperture being the same. 

Hence, convex mirrors are used as rear-view mirrors in vehicles. It is also installed behind automated teller machines as a security measure. 

The images formed by convex mirrors are always diminished, virtual and erect, irrespective of the position of the object.

Differences Between Convex Mirror and Concave Mirror

Refraction Of Light At Plane Surfaces

Introduction

Light bends while traveling from one medium to another. Since the refraction of light occurs at the surface joining two media, the refraction is a surface phenomenon.

The bending of light when it travels from one medium into another is called refraction of light.

The reason for refraction of light is the change in speed of light. 

The speed of light in an optically rarer medium is more than that in an optically denser medium. 

• Light rays passing from rarer to denser medium bends towards the normal. This makes the angle of incidence (angle between the incident ray and the normal at the point of incidence) more than that of the angle of refraction (angle between the normal and the refracted ray). 
• Light rays passing from denser to rarer medium bends away from the normal. This makes the angle of incidence (angle between the incident ray and the normal at the point of incidence) less than that of the angle of refraction (angle between the normal and the refracted ray). 
• The extent to which a light ray bends depends on the refrangibility of the ray with respect to the medium.
• In other words, the extent to which a light ray bends depends on the refractive index of the respective medium. 
• The ratio of velocity of light in vacuum to that in a medium is termed as the absolute refractive index (m) of the medium or simply termed as the refractive index of the medium. 
• Refractive index (m) of the medium is the measure of the ability of light to get bent in the given medium. 
• Measuring the speed of light is difficult.
• We can determine the refractive index using Snell’s law
• According to Snell’s law,

• The refraction of light obeys the following two laws:
  • The incident ray, the refracted ray and the normal at the point of incidence all lie in the same plane. 
  • The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant. This constant is called the index of refraction or refractive index.

Mathematically, (sin i)/(sin r)= (n₂/n₁)   (Or)

(sin i)/(sin r)= (1n₂), where (1n₂), is the refractive index of the medium 2, in which the refracted ray travels, with respect to medium 1, in which the incident ray travels. 

This law is credited to Willebrord Snell and is, therefore, called Snell’s law.

If wn_g is the refractive index of glass w.r.t. water, an_g be the refractive index of glass w.r.t. air and an_w be the refractive index of water w.r.t. air ,then

wn_g= an_g/an_w

• If the light ray retraces its path while traveling from denser to rarer, the angle of incidence is lesser than that of the refraction. This is the principle of reversibility.

The extent to which a light ray bends depends on the refrangibility of the ray with respect to the medium. 

The ratio of velocity of light in vacuum to that in a medium is termed as the absolute refractive index (m) of the medium.  Absolute refractive index (m) of the medium is the measure of the ability of light to get bent in the given medium. 

Measuring the speed of light is difficult. 

We can determine the refractive index using Snell’s law

According to Snell’s law,

Refraction of Light At A plane Surface

The bottom of a water glass appears to rise upwards when viewed normally. This is due to the vertical shift of the bottom of the glass, which takes place because of refraction.

Refraction of Light At Glass Slab

When a light ray, incident at an angle, passes through a glass slab, the emergent ray shifts laterally, known as lateral shift. 

The lateral shift depends on the thickness and refractive index of the glass slab.

If the angle of incidence increases gradually, the angle of refraction also increases. 

But, at a particular angle of incidence in the denser medium, the refracted ray emerges along the surface. That particular angle is known as the critical angle. 

If the angle of incidence is greater than the critical angle, the ray undergoes Total Internal Reflection. 

The formation of Mirages in deserts is due to total internal reflection of light.

n=cv

n=(Speed of light in vacuum)/( speed of light in the medium)

As light travels from one medium to another, the frequency of light does not change. 

Lenses

A lens is a piece of transparent optical material with one or two curved surfaces to refract light rays.

The simplest lens has two spherical surfaces close enough together that we can neglect the distance between them. Such a lens is called a thin lens. 

The two common types of lenses are Converging lens or Convex lens. 

It should be noted that, if the above lenses are surrounded by a material with a refractive index greater than that of the lens, the convex lens gets converted into a concave lens and vice versa.

A lens may converge or diverge light rays to form an image.

Types of Lenses

A bi-convex lens is one with a surface that is bulged outwards on both the sides. It is generally referred to as a convex lens.

Another type of lens is a bi-concave lens that has two inward bent surfaces. It is generally referred to as a concave lens. 

A Plano-convex lens has a convex surface on one side and a plane surface on the other.

A Plano-concave lens is the one that has a concave surface on one side and a plane surface on the other.

A concavo-convex lens has a concave surface on one side and a convex surface on the other.

Convex and concave lenses are important as they are more commonly used than the other types of lenses.

Terms Used for Lens

Center of Curvature: The center of the imaginary glass sphere of which the lens is a part, is called center of curvature.

Principal Axis: An imaginary line joining the centers of curvature of the two spheres, of which lens is a part, is called Principal Axis.

Optic Center: A point within the lens, where a line drawn through the diameter of lens meets the principal axis, is called the optic center.

Principal Focus for Convex Lens: It is a point on the principal axis of a convex lens,  where parallel beams of light rays, traveling parallel to the principal axis, after passing through the lens actually meet.

Principal Focus for Concave Lens: It is a point on the principal axis of a concave lens, from where a parallel beam of light rays, traveling parallel to the principal axis, after passing through the lens, appears to come.

Focal Length: The distance between principal focus and optical centre is called focal length.

Aperture: The effective diameter of the lens through which refraction takes place is called aperture of lens.

Optic centre is a point on the axis of a lens such that any light ray passing through this point emerges without refraction.

   •  Principal focus is a point on the axis of a lens.

   •  Principal focus is also known as the focal point.

Spherical Lenses

Convex Lense

A lens in which both the surfaces are convex, is known as convex lens.

Since the convex lens converges the light incident on it. It is called the converging lens. 

The convex lenses form different types of images depending on its relative position with respect to the position of the object in front of them. 

Thus, the magnification produced by these lenses varies.

Concave Lense

A lens in which both the surfaces are concave, is known as a concave lens.

Since the concave lens diverges the light incident on it. It is called the diverging lens. 

Due to this the concave lenses always form diminished, virtual and erect images irrespective of the position of the object in front of them. 

Thus, the magnification produced by these lenses is always less than one.

Image Formation by Refraction Of Light Through Spherical Lenses

Behaviour of Light Rays Propagating Through a Convex Lens

Rules for Construction of Ray Diagrams for Convex Lens

Rule 1:  All rays parallel to the 

principal axis of a convex lens passes through the principal focus after refraction.

Rule 2: A ray of light passing through the focus of a convex lens becomes parallel to the principal axis after refraction. 

Rule 3: A ray of light passing through the optical center of a convex lens passes un deviated after refraction.

Rule 4:

Note: A convex and a concave lens can be supposed to be made-up of prisms.

Image Formation by a Convex Lens

Concave Lens:

A lens, in which both the surfaces are concave, is known as a concave lens.

 An image formed by a concave lens is always diminished due to the divergence of rays. This is why concave lenses are widely used to correct eye defects such as myopia.

 A concave lens is also known as a diverging, reducing, negative and myopic or minus lens.

Behaviour of Light Rays Propagating Through a Concave Lens.

Behaviour of Light Rays Propagating Through a Concave Lens

Rules for Construction of Ray Diagrams for Concave Lens

Rule 1:  All  rays parallel to the principal axis of a concave lens diverge such that they are coming from the principal focus after refraction.

Rule 2: A ray of light directed towards the focus of a concave lens becomes parallel to the principal axis after refraction. 

Rule 3: A ray of light passing through the optical center of a concave lens passes un deviated.

Note: A convex and a concave lens can be supposed to be made-up of prisms.

The lens formula defines the relationship between the focal length of the lens (f), the distance of the object from the optic center (u) and the distance of the image from the optic center (v):

Location and Characteristic of the Images Formed by a Concave Lens 

Sign convention for spherical lenses:

• All distances on the principal axis are measured from the optic center of the lens.
• All distances measured above the principal axis are taken as positive. Thus, height of an object and that of an erect image are positive and all distances measured below the principal axis are taken as negative.
• The distances measured in the direction of the incident light are taken as positive (+)  
• The distances measured in the direction opposite to that of the incident light are taken as negative (-).

Table Showing Sign Convention In Concave & Convex Mirrors & Lenses

Lens Formula and Sign Conventions

Magnification of lens (m)

Magnification is the ratio of the image size to the object size. It is also measured as the ratio of image distance to object distance.

 m = Size of the image / Size of Object 

      Or 

m = Image distance / Object distance

If m = 1;  image size = object size

If m > 1:  image size > object size

If m < 1:  image size < object size

Power of Lens (P)

   •  The converging or diverging capacity of a lens is ascertained by its power

   •  Power of a lens is the reciprocal of its focal length expressed in meter. 

P=1f

 ( measured in meters).

   •  SI Unit of power of a lens is dioptre (D).

   •  Power of a convex lens is positive and that of a concave lens is negative.

Differences Between Convex Lens and Concave Lens:

Formulae:

• Magnification  = Image distance / Object distance

 m=vu

• Magnification  = Image Height  / Object Height
• Image distance / Object distance = Image Height  / Object Height
• If magnification is <1, the image is Diminished or minimised or shrunk.
• If magnification= 1, the size of the image is equal to the size of the object.
• If magnification is >1, the image is Magnified or enlarged or bigger.
• If magnification is <1, the image is Diminished or Shrunk or Smsller.
• 
• ⇒ m=vu ⇒ u=vm
• ⇒ m=vu ⇒ v=mu
• ⇒ 1f=1v1(vm) ⇒ 1f=1v-mv ⇒ 1f=1-mv
• ⇒ f=v1-m
• ⇒ f=mu1-m
• ⇒ Power of lens=1Focal Length
• ⇒ Power of lens x Focal Length =1
• ⇒ P=1f

For Spherical Mirrors :

P= 2R=1f=1v+ 1u=1-mv =(m-1)mu-(m-1)-mu

For Spherical Lenses :

P= 2R=1f= 1v-1u=1-mv = 1-mmu

• For Spherical Mirrors :
• R2=f= uvu+v
• 2R=1f= 1v + 1u
• m=-vu
• If m > 1, Image is magnified
• If m < 1, Image is diminished
• If m = 1, Image is Equal to size of object
• Sign of magnification, m is negative for real image.
• Sign of magnification, m is positive for virtual images.
• The number indicates how many times magnification.
• Where the sign indicates whether the image is virtual erect or real inverted.
• v=-mu
• m=hih0

• For Spherical Lenses :
• R2=f= uvu-v
• 2R=1f= 1v – 1u
• m=vu
• m=hih0
• If m > 1, Image is magnified
• If m < 1, Image is diminished
• If m = 1, Image is Equal to size of object
• Sign of magnification, m is negative for a real, Inverted image.
• Sign of magnification, m is positive for virtual, Erect image.
• The number indicates how many times magnification.
• Where the sign indicates whether the image is virtual erect or real inverted.

• P= 2R=1-mv = 1-mmu

• 1f= 1v+1u
• If m is Negative image is real
• If m is Positive image is Virtual
• If height of the image is equal to height of the object m = 1, that is image height is equal to the object height
• If height of the image is greater than height of the object, then m > 1. That is, the image is enlarged.
• If height of the image is less than height of the object then m<1, That is, the image is diminished.
• Magnification of a Plane mirror is always +1.
• Positive Sign of the magnification indicates Virtual, erect image.
• Negative Sign of the magnification indicates Real , Inverted image.