Light - Reflection and Refraction — Notes

Reflection of Light

Reflection is the phenomenon where light bounces off a surface. When light encounters a highly polished surface, like that of a mirror, most of it bounces back into the same medium. This is how we see objects: light from a source reflects off the object and then reflects into our eyes.

The fundamental principles governing reflection are known as the Laws of Reflection:

1. The angle of incidence is equal to the angle of reflection.
2. The incident ray, the normal to the surface at the point of incidence, and the reflected ray all lie in the same plane.

These laws are universal and apply to all types of reflecting surfaces, including flat (plane) surfaces and curved (spherical) surfaces.

Plane Mirrors and Image Formation

You are familiar with images formed by a plane mirror. These images have specific properties:

• Virtual: The image cannot be projected onto a screen. It appears to be behind the mirror.
• Erect: The image is upright, not upside down.
• Same Size: The size of the image is identical to the size of the object.
• Equidistant: The image is formed as far behind the mirror as the object is in front of it. For an object at distance $d$ from the mirror, the image is also at distance $d$ behind the mirror.
• Laterally Inverted: The image is reversed left-to-right. For example, if you raise your right hand, the image appears to raise its left hand.

Curved Reflecting Surfaces

When the reflecting surface is not flat but curved, the nature of the reflected image changes. This can be observed using a shining spoon.

• The inner curved surface of the spoon acts like a concave mirror, and the outer curved surface acts like a convex mirror.
• These curved mirrors can form images that are magnified, diminished, or of the same size as the object.
• The images formed can be erect or inverted, and also virtual or real (a real image can be projected onto a screen).

The behavior of light reflecting from curved surfaces is crucial for understanding optical instruments like telescopes and microscopes, and even the design of headlights and rearview mirrors.

Remember

The laws of reflection ($angl{e}_i=angl{e}_r$) are the bedrock for all reflection phenomena, whether from plane or spherical surfaces.

Exam-smart

Case study questions might describe a scenario with an object and a polished, curved surface (like a spoon). You'll need to identify whether it's acting as a concave or convex mirror and predict basic image properties (erect/inverted, magnified/diminished) based on the object's position relative to the surface.

Spherical Mirrors

A spherical mirror is a mirror whose reflecting surface is a part of the surface of a sphere. These mirrors are used widely due to their ability to focus light.

Types of Spherical Mirrors

Spherical mirrors are classified based on the curvature of their reflecting surface:

A shining spoon can serve as an example: the inner, curved surface acts like a concave mirror, and the outer, bulging surface acts like a convex mirror.

Key Terms Associated with Spherical Mirrors

To understand the behavior of spherical mirrors, several terms are essential:

• Pole (P): The geometric center of the reflecting surface of a spherical mirror. It lies on the surface of the mirror.
• Center of Curvature (C): The center of the sphere from which the reflecting surface of the mirror is a part. It does not lie on the mirror itself but outside its reflecting surface. For a concave mirror, C is in front; for a convex mirror, C is behind.
• Radius of Curvature (R): The radius of the sphere of which the reflecting surface is a part. It is the distance from the pole (P) to the center of curvature (C).
• Principal Axis: An imaginary straight line passing through the pole (P) and the center of curvature (C) of the spherical mirror. It is normal to the mirror at its pole.
• Principal Focus (F):
• Concave Mirror: The point on the principal axis where parallel rays of light incident on the mirror converge after reflection.
• Convex Mirror: The point on the principal axis from which parallel rays of light, after reflection, appear to diverge.
• Focal Length (f): The distance between the pole (P) and the principal focus (F) of a spherical mirror. For mirrors with small apertures, the principal focus lies midway between the pole and the center of curvature.
• Aperture: The diameter of the reflecting surface of a spherical mirror. For simplicity, we consider mirrors where the aperture is much smaller than the radius of curvature.

Relationship between Focal Length and Radius of Curvature

For spherical mirrors with small apertures, the focal length is exactly half of the radius of curvature:

This implies that the radius of curvature is twice the focal length:

Image Formation by Spherical Mirrors

The nature, position, and size of the image formed by spherical mirrors depend on the position of the object relative to the pole (P), principal focus (F), and center of curvature (C).

Image Formation by Concave Mirrors:

Concave mirrors can form both real (formed on the same side as the object, can be projected on a screen) and virtual (formed behind the mirror, cannot be projected on a screen) images. The image characteristics vary with the object's position:

Image Formation by Convex Mirrors:

Convex mirrors always form virtual, erect, and diminished images, regardless of the object's position. The image is always formed behind the mirror, between the pole (P) and the principal focus (F).

Uses of Spherical Mirrors:

• Concave Mirrors:
• Used in torches, searchlights, and vehicle headlights to produce powerful, parallel beams of light.
• Used as shaving mirrors or dentists' mirrors to obtain a larger, erect image of the face or teeth.
• Used in solar furnaces to concentrate sunlight for heating.
• Convex Mirrors:
• Commonly used as rear-view (wing) mirrors in vehicles. They provide a wider field of view, allowing the driver to see more of the traffic behind, and always show an erect image.

Representation of Images Using Ray Diagrams

Ray diagrams help visualize image formation. We use specific rays whose reflection is predictable:

1. A ray parallel to the principal axis reflects through (concave) or appears to come from (convex) the principal focus (F).
2. A ray passing through the principal focus (F) reflects parallel to the principal axis.
3. A ray passing through the center of curvature (C) reflects back along the same path.
4. A ray incident on the pole (P) reflects obeying the laws of reflection (angle of incidence = angle of reflection).

The intersection of at least two reflected rays determines the position of the image.

Sign Convention for Reflection by Spherical Mirrors (New Cartesian Sign Convention)

To solve numerical problems, a consistent sign convention is used:

Mirror Formula and Magnification

• Mirror Formula: Relates object distance ($u$), image distance ($v$), and focal length ($f$):

This formula is valid for all spherical mirrors and all positions of the object when the New Cartesian Sign Convention is applied.

• Magnification (m): The ratio of the height of the image ($h^{\mathrm{\prime }}$) to the height of the object ($h$). It indicates how much the image is magnified or reduced relative to the object.

Magnification is also related to object and image distances:

• If $m$ is negative, the image is real and inverted.
• If $m$ is positive, the image is virtual and erect.
• If $\mathrm{\mid }m\mathrm{\mid }>1$, the image is enlarged.
• If $\mathrm{\mid }m\mathrm{\mid }<1$, the image is diminished.
• If $\mathrm{\mid }m\mathrm{\mid }=1$, the image is of the same size as the object.

Remember

For spherical mirrors, the focal length is typically half the radius of curvature ($f=R\mathrm{/}2$). Always apply the New Cartesian Sign Convention when substituting values into the mirror formula and magnification equation.

Exam-smart

Case study questions often involve applying the mirror formula and magnification to determine image characteristics (position, nature, size) for a given object placed before a specific type of spherical mirror. You might be given the radius of curvature or focal length and object position, and asked to find where the image forms or its magnification.

Refraction of Light

Refraction is the phenomenon where light changes its direction when it travels obliquely from one transparent medium to another. This happens because the speed of light changes as it enters a different medium. We observe this effect in everyday situations, such as a pencil appearing bent in water or objects underwater looking raised.

Why does refraction happen? Light travels in straight lines in a uniform medium. However, when it crosses the boundary between two different transparent media (like air and water, or glass and air), its speed changes. This change in speed causes the light ray to bend. The amount of bending depends on the pair of media involved.

Real-life examples of refraction:

• Apparent raising of underwater objects: The bottom of a pond or tank appears shallower than it actually is.
• Bent appearance of a pencil in water: A pencil partly immersed in water seems to be bent at the water's surface.
• Larger appearance of objects in water: A lemon in a tumbler of water appears bigger when viewed from the sides.

Refraction through a Rectangular Glass Slab

When light passes through a rectangular glass slab, it undergoes refraction twice: once when entering the glass from air, and again when exiting the glass into air.

• Incident ray: The ray of light that falls on the surface of the glass slab.
• Refracted ray: The ray of light as it passes through the glass slab.
• Emergent ray: The ray of light as it leaves the glass slab.

Observations:

1. When a light ray enters the glass slab obliquely (not perpendicular to the surface), it bends towards the normal at the air-glass interface. This is because glass is optically denser than air, and light slows down.
2. When the light ray exits the glass slab into air, it bends away from the normal at the glass-air interface. This is because air is optically rarer than glass, and light speeds up.
3. The emergent ray is parallel to the direction of the incident ray. Although the direction changes inside the slab, the overall lateral shift is caused by the parallel nature of the slab's surfaces.
4. If the light ray is incident normally (perpendicular to the surface), it does not bend and passes straight through.

Remember

The bending of light occurs because of the change in its speed. Bending towards the normal happens when light enters a denser medium, and away from the normal when it enters a rarer medium.

Exam-smart

A diagram of a rectangular glass slab showing incident, refracted, and emergent rays will often be provided. You'll be asked to identify the angles of incidence and refraction, and to explain why the emergent ray is parallel to the incident ray despite a lateral shift.

Laws of Refraction of Light

The behavior of light during refraction is governed by two fundamental laws:

1. The first law states that the incident ray, the refracted ray, and the normal to the interface of the two transparent media at the point of incidence, all lie in the same plane.
2. The second law, also known as Snell's Law, states that for a given pair of media and a given color of light, the ratio of the sine of the angle of incidence ($i$) to the sine of the angle of refraction ($r$) is a constant.

This can be expressed mathematically as:

This constant is called the refractive index of the second medium with respect to the first medium.

Remember

Snell's Law only applies when light enters a medium obliquely. For normal incidence ($i=0$), the angle of refraction ($r$) is also 0, and the law doesn't apply in its ratio form.

Exam-smart

You might be given angles of incidence and refraction and asked to calculate the sine ratio, or vice versa. Understanding the relationship between these angles is crucial.

The Refractive Index

The refractive index ($n$) of a medium quantifies how much light slows down and bends when entering that medium from another. It is a measure of the optical density of the medium.

• A higher refractive index means light travels slower in that medium and bends more.
• A lower refractive index means light travels faster and bends less.

The refractive index of the second medium with respect to the first medium ($n_{21}$) is given by:

When light travels from a vacuum (or air, which has a very similar refractive index to vacuum) into another medium, the refractive index is called the absolute refractive index of that medium. It is denoted by $n$. The speed of light in vacuum is denoted by $c$.

Where:

• $c$ is the speed of light in vacuum ($\approx 3\times {10}^8$ m/s).
• $v$ is the speed of light in the medium.

Since the speed of light in any medium is less than its speed in vacuum, the refractive index of any medium is always greater than 1. The refractive index of air is approximately 1.0003, so it is often considered as 1 for practical purposes.

Refractive Index Values for Some Media:

Remember

A medium with a higher refractive index is called optically denser, and a medium with a lower refractive index is called optically rarer. Light bends towards the normal when going from rarer to denser, and away from the normal when going from denser to rarer.

Exam-smart

You might be given the refractive indices of two media and asked to determine which one is optically denser, or to compare how light bends in each. You could also be given the refractive index and asked to calculate the speed of light in that medium.

The Refractive Index

The refractive index ($n$) of a medium quantifies how much light slows down when passing through it compared to its speed in a vacuum. It is defined as the ratio of the speed of light in a vacuum ($c$) to the speed of light in the medium ($v$).

Key Points:

• Light travels fastest in a vacuum ($c=3\times {10}^8$ m/s).
• The speed of light in air is only slightly less than in a vacuum.
• In denser media like water or glass, the speed of light is significantly reduced.
• A higher refractive index means light travels slower in that medium, making it optically denser.
• A lower refractive index means light travels faster, making it optically rarer.

When light travels from one medium (medium 1) to another (medium 2) obliquely, it changes direction. The refractive index of medium 2 with respect to medium 1 ($n_{21}$) is given by:

If medium 1 is a vacuum or air, the refractive index of medium 2 is called the absolute refractive index ($n$) of that medium.

Optical Density vs. Mass Density: Optical density is related to the refractive index, not mass density. A medium with a higher refractive index is optically denser, even if its mass density is lower. For example, kerosene is optically denser than water.

• Light travelling from an optically rarer medium to an optically denser medium bends towards the normal.
• Light travelling from an optically denser medium to an optically rarer medium bends away from the normal.

Example Application:

• Snell's Law (from previous section): $\frac{\sin i}{\sin r}=n_{21}$, which relates the angles of incidence and refraction to the refractive index. This law quantitatively describes how light bends.

Remember

The refractive index is always greater than or equal to 1, as the speed of light in any medium is less than or equal to its speed in a vacuum.

Exam-smart

You might be given the speed of light in two media and asked to calculate the refractive index, or given the refractive index and the speed in one medium to find the speed in another. You could also be asked to compare the speeds of light in different materials based on their refractive indices.

Refraction by Spherical Lenses

A lens is a transparent material, usually made of glass or plastic, bounded by at least one and often two spherical surfaces. Lenses are used to converge or diverge light rays, making them essential components in many optical instruments.

• Convex Lens: Thicker in the middle than at the edges. It converges parallel rays of light to a point. Also known as a converging lens.
• Concave Lens: Thicker at the edges than in the middle. It diverges parallel rays of light, making them appear to originate from a point. Also known as a diverging lens.

Key Terms for Lenses:

• Centre of Curvature (C): The center of the sphere from which a lens surface is a part. A lens has two centres of curvature ($C_1$ and $C_2$).
• Principal Axis: An imaginary line passing through the two centres of curvature.
• Optical Centre (O): The central point of a lens. A ray passing through the optical centre goes undeviated.
• Aperture: The effective diameter of the circular outline of a spherical lens.
• Principal Focus (F): For a convex lens, it's the point on the principal axis where parallel rays converge after refraction. For a concave lens, it's the point from which parallel rays appear to diverge after refraction. Lenses have two principal foci ($F_1$ and $F_2$).
• Focal Length (f): The distance from the optical centre to the principal focus.

Image Formation by Lenses

Lenses form images by refracting light. The nature, position, and size of the image depend on the type of lens and the object's position.

Image Formation by a Convex Lens:

Image Formation by a Concave Lens:

A concave lens always forms a virtual, erect, and diminished image, irrespective of the object's position. The image is formed between the optical centre (O) and the principal focus (F) on the same side as the object.

Remember

A convex lens can form both real and virtual images, while a concave lens always forms virtual images.

Exam-smart

Case studies often involve identifying the type of lens based on image characteristics (real/virtual, erect/inverted, magnified/diminished) or determining the object position required for a specific image.

Image Formation in Lenses Using Ray Diagrams

Ray diagrams help visualize image formation. Three principal rays are used for lenses:

1. A ray parallel to the principal axis, after refraction, passes through (convex lens) or appears to diverge from (concave lens) the principal focus ($F$).
2. A ray passing through the principal focus ($F$), after refraction, emerges parallel to the principal axis.
3. A ray passing through the optical centre ($O$) emerges without any deviation.

Sign Convention for Spherical Lenses

Similar to spherical mirrors, a New Cartesian Sign Convention is used for lenses:

• All distances are measured from the optical centre (O) of the lens.
• The object is placed to the left of the lens.
• Distances measured in the direction of incident light are positive.
• Distances measured in the opposite direction of incident light are negative.
• Distances measured upwards from the principal axis are positive.
• Distances measured downwards from the principal axis are negative.

Conventions for Focal Length:

• Convex Lens: Focal length ($f$) is positive.
• Concave Lens: Focal length ($f$) is negative.

Lens Formula and Magnification

The lens formula relates the object distance ($u$), image distance ($v$), and focal length ($f$) for any spherical lens:

Magnification (m): Magnification is the ratio of the height of the image ($h^{\mathrm{\prime }}$) to the height of the object ($h$). It also relates to the image and object distances.

• A positive magnification indicates a virtual and erect image.
• A negative magnification indicates a real and inverted image.
• $\mathrm{\mid }m\mathrm{\mid }>1$: Magnified image.
• $\mathrm{\mid }m\mathrm{\mid }<1$: Diminished image.
• $\mathrm{\mid }m\mathrm{\mid }=1$: Image is the same size as the object.

Remember

The lens formula sign is a minus (-) in the middle, unlike the mirror formula's plus (+). Be consistent with the sign convention.

Exam-smart

Problems will often provide two of the values (object distance, image distance, focal length) and ask for the third, or ask for magnification given distances or heights. You must correctly apply the sign convention.

Example Calculations:

• Example 9.3 (Concave lens): Focal length $f=-15$ cm, image distance $v=-10$ cm. Using the lens formula to find object distance $u=-30$ cm. Magnification $m=v\mathrm{/}u=-10\mathrm{/}-30=+0.33$. This positive magnification confirms a virtual, erect, and diminished image.
• Example 9.4 (Convex lens): Object height $h=+2.0$ cm, focal length $f=+10$ cm, object distance $u=-15$ cm. Using the lens formula to find image distance $v=+30$ cm. Magnification $m=v\mathrm{/}u=+30\mathrm{/}-15=-2$. Height of image $h^{\mathrm{\prime }}=m\times h=-2\times 2.0=-4.0$ cm. This negative magnification and image height indicate a real, inverted, and enlarged image.

Power of a Lens

The power of a lens (P) is a measure of its ability to converge or diverge light. It is defined as the reciprocal of its focal length ($f$) when the focal length is expressed in meters.

• The SI unit of power is the dioptre (D).
• 1 dioptre is the power of a lens with a focal length of 1 metre.

Key Points:

• Convex Lens: Has a positive focal length, hence positive power.
• Concave Lens: Has a negative focal length, hence negative power.

Combination of Lenses: When several lenses are placed in contact, the net power of the combination is the algebraic sum of the individual powers:

This principle is very useful for opticians to prescribe corrective lenses for vision defects.

Remember

Always express focal length in metres before calculating power in dioptres. If focal length is given in cm, convert it to metres by dividing by 100.

Exam-smart

You might be given the focal length of a lens and asked to find its power, or vice versa. You could also be asked to find the power of a combination of lenses or determine the type of lens based on its power. For example, a power of +2.0 D indicates a converging (convex) lens, while -2.5 D indicates a diverging (concave) lens.