Rays and Beams

A ray of light represents the direction in which light travels. A beam is a bundle of many light rays traveling together.

Rectilinear Propagation

Light travels in straight lines in a uniform medium. This can be demonstrated by making a small hole in cardboards arranged in a straight line – when you look through the last hole, you can see a candle flame only when all holes are aligned in a straight line.

Transmission of Light

Materials can be classified according to how they transmit light:

Shadows

When an opaque object blocks light, a shadow is formed. The shadow has two parts:

• Umbra – The dark central region where light is completely blocked
• Penumbra – The lighter region around the umbra where only part of the light is blocked

Point sources (small light sources) produce sharp shadows, while extended sources (like the Sun) produce shadows with both umbra and penumbra.

Reflection occurs when light bounces back from a surface. There are two types:

Regular Reflection

Occurs on smooth, polished surfaces like mirrors. All reflected rays are parallel, producing clear images.

Irregular (Diffuse) Reflection

Occurs on rough surfaces. Reflected rays scatter in different directions, so no clear image is formed.

Laws of Reflection

1. The angle of incidence ($i$) equals the angle of reflection ($r$).
2. The incident ray, reflected ray, and the normal to the surface all lie in the same plane.

$i = r$

A plane mirror forms images with the following characteristics:

• Virtual – The image appears to be behind the mirror; it cannot be projected on a screen
• Upright – The image is the same orientation as the object
• Same size – Magnification is 1 (image height equals object height)
• Same distance – The image is the same distance behind the mirror as the object is in front
• Laterally inverted – The left side of the object appears as the right side of the image (and vice versa)

Worked Example

An object is placed 2 cm from a plane mirror. If the object is moved 1 cm toward the mirror, what is the new distance between the object and its image?

Solution:

Initial distance from mirror = 2 cm

After moving 1 cm toward mirror: new distance = 2 cm − 1 cm = 1 cm

In a plane mirror, image distance = object distance from mirror

Therefore, image distance = 1 cm behind the mirror

Total distance between object and image = 1 cm + 1 cm = 2 cm

Types of Curved Mirrors

Concave mirror: The reflecting surface curves inward (like the inside of a spoon). It can produce real or virtual images depending on object position.

Convex mirror: The reflecting surface curves outward. It always produces virtual, diminished, and upright images.

Key Terms

• Principal axis: The line passing through the center of curvature and the pole (center) of the mirror
• Centre of curvature (C): The center of the sphere from which the mirror is part
• Pole (P): The center of the mirror surface
• Focal point (F): The point where parallel rays converge (concave) or appear to diverge from (convex)
• Focal length (f): Distance from pole to focal point; $f = R/2$ where R is the radius of curvature

Image Formation by Concave Mirrors

Image Formation by Convex Mirrors

Convex mirrors always produce virtual, upright, and diminished images located between the focal point and the mirror.

Mirror Formula

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

Where:

• $u$ = object distance (positive if object is in front of mirror)
• $v$ = image distance (positive for real images, negative for virtual)
• $f$ = focal length (positive for concave, negative for convex)

Magnification

$m = \frac{\text{image height}}{\text{object height}} = \frac{v}{u}$

Or equivalently:

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

The negative sign indicates image inversion.

Worked Example

An object 5 cm high is placed 10 cm from a concave mirror of focal length 15 cm. Find the image distance and height.

Given: $h_o = 5$ cm, $u = 10$ cm, $f = 15$ cm

Using mirror formula: $\frac{1}{v} = \frac{1}{f} - \frac{1}{u} = \frac{1}{15} - \frac{1}{10} = \frac{2-3}{30} = -\frac{1}{30}$

Therefore: $v = -30$ cm (virtual image, behind the mirror)

Now for magnification: $m = -\frac{v}{u} = -\frac{(-30)}{10} = 3$

Image height: $h_i = m \times h_o = 3 \times 5 = 15$ cm

The image is virtual, upright, and enlarged (3 times the object size).

Refraction is the bending of light as it passes from one medium to another with different optical density.

Laws of Refraction (Snell's Law)

1. The incident ray, refracted ray, and normal all lie in the same plane.
2. The ratio of sine of angle of incidence to sine of angle of refraction is constant: $\frac{\sin i}{\sin r} = \eta$

Where $\eta$ is the refractive index of the material.

Refractive Index

$\eta = \frac{c}{v}$

Where $c$ is speed of light in vacuum ($3 \times 10^8$ m/s) and $v$ is speed of light in the material.

Or for relative refractive index: $\eta_{12} = \frac{\sin i}{\sin r} = \frac{v_1}{v_2}$

Effects of Refraction

• Light going from less dense to denser medium bends toward the normal.
• Light going from denser to less dense medium bends away from the normal.
• When light enters at 90° (normal incidence), it continues without bending.

Total Internal Reflection

This occurs when:

1. Light travels from a denser to a less dense medium
2. Angle of incidence exceeds the critical angle

The critical angle is given by: $\sin c = \frac{1}{\eta}$

Types of Lenses

Convex (converging) lens: Thicker in the middle; brings parallel rays to a focus. Used for correcting long-sightedness.

Concave (diverging) lens: Thinner in the middle; makes parallel rays diverge. Used for correcting short-sightedness.

Lens Formula

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

Sign convention: Distances measured against the direction of incident light are negative.

Image Formation by Convex Lens

Image Formation by Concave Lens

Concave lenses always produce virtual, upright, and diminished images.

Simple Microscope (Magnifying Glass)

A simple microscope is a convex lens used to produce a magnified virtual image of a small object. Angular magnification: $M = 1 + \frac{D}{f}$

Where $D \approx 25$ cm (least distance of distinct vision) and $f$ is focal length.

Compound Microscope

Uses two convex lenses (objective and eyepiece) to achieve higher magnification: $M = \frac{L}{f_o} \times \frac{D}{f_e}$

Where $L$ is tube length, $f_o$ is objective focal length, $f_e$ is eyepiece focal length.

Astronomical Telescope

Uses two convex lenses to view distant objects. Angular magnification: $M = \frac{f_o}{f_e}$

Where $f_o$ is objective focal length and $f_e$ is eyepiece focal length.

The Human Eye

The eye works like a camera:

• Cornea: Provides most of the focusing power
• Lens: Fine adjustments for accommodation (changing focus for different distances)
• Iris: Controls pupil size (like camera diaphragm)
• Retina: Where image is formed (like film/sensor)
• Near point: About 25 cm for normal vision
• Far point: Infinity for normal vision

Common Vision Defects

In Tanzania, farmers use the principle of light refraction when drying crops like maize and cassava in the sun – understanding that transparent plastics (like greenhouse covers used in some modern Tanzanian horticulture) allow light to pass through and convert to heat helps in designing better drying and farming structures. Additionally, mirrors are commonly used in small businesses like beauty salons in Dar es Salaam and other towns, where the principle of reflection allows customers to see themselves clearly – salon owners often arrange multiple mirrors at angles to provide wider views using the concept of multiple reflections taught in this topic.