10. Light - Reflection and Refraction

1. Introduction to Light and its Nature

Sunlight or artificial light reflects off objects. When this reflected light is received by our eyes, it enables us to see. Transparent media transmit light, allowing us to see through them.

• Straight-line Propagation: Common phenomena like sharp shadows cast by opaque objects indicate that light generally travels in straight lines. A straight-line path is called a ray of light.

• Wave vs. Particle Dilemma:

  • Diffraction: If an opaque object on the path of light becomes extremely small, light tends to bend around it rather than walking in a straight line. To explain this, light is treated as a wave.

  • Stream of Particles: At the start of the 20th century, wave theory proved inadequate for explaining light's interaction with matter, making it behave like a stream of particles.

  • Modern Quantum Theory: Reconciles both views, stating light is neither purely a wave nor a particle.

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2. Reflection of Light

Reflection occurs when a highly polished surface (like a mirror) bounces back most of the light falling on it.

Laws of Reflection

Applicable to all types of reflecting surfaces, including spherical surfaces:

1. The angle of incidence is equal to the angle of reflection ($i = r$).

2. The incident ray, the normal to the mirror at the point of incidence, and the reflected ray all lie in the same plane.

Plane Mirror Characteristics

• Image is always virtual and erect.

• The size of the image is equal to the size of the object.

• The image is as far behind the mirror as the object is in front of it.

• The image is laterally inverted.

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3. Spherical Mirrors

Mirrors whose reflecting surfaces are spherical are called spherical mirrors. They are parts of a sphere.

Types of Spherical Mirrors

• Concave Mirror: Reflecting surface is curved inwards (faces the centre of the sphere). It acts as a converging mirror.

• Convex Mirror: Reflecting surface is curved outwards. It acts as a diverging mirror.

Important Terms

• Pole (P): The centre of the reflecting surface of the mirror, lying on its surface.

• Centre of Curvature (C): The centre of the sphere of which the mirror forms a part. It lies outside the reflecting surface (in front of a concave mirror, behind a convex mirror).

• Radius of Curvature (R): The radius of the sphere of which the mirror forms a part; the distance $PC = R$.

• Principal Axis: The straight line passing through the pole ($P$) and the centre of curvature ($C$). It is normal to the mirror at its pole.

• Principal Focus (F):

  • For a concave mirror, it is the point on the principal axis where rays parallel to the axis intersect after reflection.

  • For a convex mirror, it is the point from which parallel rays appear to diverge after reflection.

• Focal Length (f): The distance between the pole ($P$) and the principal focus ($F$).

• Relationship: For mirrors with small apertures, the focus lies midway between $P$ and $C$:

  $R = 2f$

• Aperture: The diameter of the circular outline of the reflecting surface.

4. Image Formation by Spherical Mirrors

To locate an image using a ray diagram, we track the intersection of at least two standard rays:

1. A ray parallel to the principal axis passes through $F$ (concave) or appears to diverge from $F$ (convex).

2. A ray passing through or directed towards $F$ emerges parallel to the principal axis.

3. A ray passing through or directed towards $C$ reflects back along its own path because it strikes the mirror normally.

4. A ray incident obliquely at the pole $P$ reflects obliquely, obeying $i = r$ relative to the principal axis.

Summary Tables of Image Formation

Concave Mirror

Convex Mirror

Applications/Uses

• Concave Mirrors: Used in torches, search-lights, and vehicle headlights for parallel beams ; as shaving mirrors ; by dentists to see enlarged teeth ; and in solar furnaces to concentrate sunlight.

• Convex Mirrors: Used as rear-view (wing) mirrors in vehicles because they always provide an erect, diminished image and offer a wider field of view since they bulge outwards.

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5. Mirror Formula and Magnification

New Cartesian Sign Convention

• The pole ($P$) is treated as the origin; the principal axis is the x-axis ($X'X$).

• Objects are always placed on the left (light falls from left to right).

• Distances measured to the left of the origin are negative; to the right are positive.

• Heights measured perpendicular above the principal axis are positive; below are negative.

• Hence: Focal length ($f$) of a concave mirror is always negative; $f$ of a convex mirror is always positive.

Mirror Formula

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

(where $u$ = object distance, $v$ = image distance, $f$ = focal length).

Magnification ($m$)

$m = \frac{\text{Height of image } (h')}{\text{Height of object } (h)} = -\frac{v}{u}$

• A negative $m$ indicates a real image.

• A positive $m$ indicates a virtual image.

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6. Refraction of Light

Refraction is the change in the direction of propagation of light when it travels obliquely from one transparent medium to another. It is caused by the change in the speed of light between different media.

• Real-life examples: A pond appearing shallower than it is ; letters appearing raised under a glass slab ; a pencil appearing bent/displaced when partially dipped in water.

Laws of Refraction

1. The incident ray, the refracted ray, and the normal to the interface at the point of incidence all lie in the same plane.

2. Snell’s Law: The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant for light of a given colour and a given pair of media:

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

Refraction Through a Rectangular Glass Slab

When light passes through a parallel-faced glass slab, it bends towards the normal entering from air to glass (rarer to denser) , and bends away from the normal exiting from glass to air (denser to rarer).

• The overall face-to-face bending is equal and opposite, so the emergent ray is parallel to the incident ray but suffers a slight sideward shift (lateral displacement).

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7. Refractive Index ($n$)

The refractive index represents the extent to which a pair of media changes the speed of light.

Relative Refractive Index

The refractive index of medium 2 with respect to medium 1 ($n_{21}$) is the ratio of the speed of light in medium 1 ($v_1$) to that in medium 2 ($v_2$):

$n_{21} = \frac{v_1}{v_2}$

Absolute Refractive Index ($n_m$)

When medium 1 is vacuum or air, the refractive index of medium 2 is its absolute refractive index:

$n_m = \frac{\text{Speed of light in air/vacuum } (c)}{\text{Speed of light in the medium } (v)}$

(Speed of light in vacuum/air $c \approx 3 \times 10^8 \text{ m s}^{-1}$).

Optical Density

• Optically Denser Medium: The medium with the higher refractive index where light slows down and bends towards the normal.

• Optically Rarer Medium: The medium with the lower refractive index where light speeds up and bends away from the normal.

• Note: Optical density is different from mass density. For example, kerosene has a higher refractive index than water (is optically denser), yet its mass density is lower than water.

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8. Refraction by Spherical Lenses

A lens is a piece of transparent material bound by two surfaces, of which one or both are spherical.

Types of Lenses

• Convex Lens (Double Convex): Thicker in the middle than at the edges. It converges parallel rays of light to a point called the principal focus ($F_2$), hence it is known as a converging lens.

• Concave Lens (Double Concave): Thicker at the edges than in the middle. It diverges parallel light rays so they appear to originate from a principal focus ($F_1$), hence it is known as a diverging lens.

Basic Terms

• Optical Centre (O): The central point of a lens. A ray passing through $O$ suffers no deviation.

• Centres of Curvature ($C_1, C_2$): Lenses have two spherical surfaces, each forming part of a sphere with its own centre.

• Principal Axis: The imaginary straight line passing through both centres of curvature.

• Aperture: The effective diameter of the circular outline of the lens.

9. Image Formation by Lenses

Standard rays used to plot lens image layouts:

1. A ray parallel to the principal axis refracts through the focus $F_2$ (convex) or appears to diverge from $F_1$ (concave).

2. A ray passing through $F_1$ (convex) or directed towards $F_2$ (concave) emerges parallel to the principal axis.

3. A ray passing through the optical centre $O$ emerges completely undeviated.

Summary Tables of Image Formation

Convex Lens

Concave Lens

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10. Lens Formula, Magnification, and Power

Sign Convention for Lenses

Similar to mirrors, but all distances are measured from the optical centre ($O$).

• Focal length ($f$) of a convex lens is always positive.

• Focal length ($f$) of a concave lens is always negative.

Lens Formula

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

Magnification ($m$)

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

Power of a Lens ($P$)

Power measures the degree of convergence or divergence a lens introduces to light rays. It is defined as the reciprocal of its focal length expressed in metres:

$P = \frac{1}{f \text{ (in meters)}}$

• SI Unit: dioptre (D). $1\text{ D} = 1\text{ m}^{-1}$.

• Power of a convex lens is positive; power of a concave lens is negative.

• Combination of Lenses: When multiple lenses are in close contact, the net optical power is the simple algebraic sum of individual powers:

  $$P = P_1 + P_2 + P_3 + \do