Refraction of Light at Plane Surfaces notes: ICSE Class 10 Physics (Science Paper 1)

1. Refraction and its laws

Refraction: the change in direction of light as it passes obliquely from one transparent medium into another. Cause: light has different speeds in different media.
At every boundary there is also partial reflection (a faint reflection in a window pane).
Rarer → denser: ray bends towards the normal; denser → rarer: away from the normal.
Laws of refraction: (1) incident ray, refracted ray and normal at the point of incidence lie in one plane; (2) Snell’s law — $\sin i / \sin r$ is constant for a given pair of media and a given colour.
On entering a denser medium: speed and wavelength decrease, frequency (and so colour) is unchanged.
No deviation when (i) $i = 0$ (along the normal) or (ii) both media have the same refractive index.

Table of the changes in light on entering a denser medium
Into a denser medium	Change
Speed	Decreases
Wavelength	Decreases
Frequency	Unchanged
Direction (oblique ray)	Bends towards the normal

Formulas

Snell’s law
$\frac{\sin i}{\sin r} = {}_{1}\mu_{2}$
Wave equation
$v = f \lambda$

Key terms

Refraction: Bending of light when it passes obliquely from one transparent medium into another, because its speed changes.
Snell’s law: For a given pair of media and colour of light, the sine of the angle of incidence divided by the sine of the angle of refraction is a constant.

2. Refractive index

Absolute refractive index: $\mu = c / v$ , with $c = 3 \times 10^{8}$ m s⁻¹. No unit; greater than 1 for every material medium.
Larger $\mu$ → slower light → optically denser (optical density is not the same as mass density: kerosene is optically denser than water but floats on it).
Relative refractive index: $\,{}_{1}\mu_{2} = \mu_{2} / \mu_{1} = v_{1} / v_{2}$ .
$\mu$ depends on the material, the colour (wavelength) of light (larger for violet than red) and slightly on temperature.
Typical values: water 1.33, glass 1.5, diamond 2.42.

Table of refractive indices and speeds of light in three media
Medium	μ	Speed (× 10⁸ m/s)
Water	1.33	2.25
Glass	1.5	2.0
Diamond	2.42	1.24

Formulas

Refractive index from speeds
$\mu = \frac{c}{v}$
Relative refractive index
${}_{1}\mu_{2} = \frac{\mu_{2}}{\mu_{1}} = \frac{v_{1}}{v_{2}}$

Key terms

Absolute refractive index: Ratio of the speed of light in vacuum to the speed of light in the medium.
Optically denser medium: A medium with a larger refractive index, in which light travels more slowly.

3. Rectangular glass block and thick mirror

Ray bends towards the normal at the first face and away at the second; the faces are parallel, so e = i and the emergent ray is parallel to the incident ray.
Lateral displacement: perpendicular distance between the incident ray produced and the emergent ray. It increases with the thickness of the block, the angle of incidence and the refractive index; it is zero for normal incidence.
Thick plane mirror (silvered at the back): partial reflection at the front glass surface gives a faint first image; reflection at the silvered back gives the brightest (second) image; repeated internal reflections give further, fainter images. All emergent rays are parallel.

Table of the images formed by a thick plane mirror
Image in a thick mirror	Formed by	Brightness
First	Reflection at the front glass surface	Faint
Second	Reflection at the silvered back	Brightest
Third and later	Repeated internal reflection	Fainter and fainter

Formulas

Parallel-sided block
$e = i$

Key terms

Lateral displacement: The perpendicular distance between the incident ray produced and the emergent ray of a parallel-sided block.
Angle of emergence: The angle between the emergent ray and the normal at the second face.

4. Real and apparent depth

Rays from an object under water bend away from the normal on leaving; produced back they meet above the object, which appears raised.
For near-normal viewing: $\mu = \dfrac{\text{real depth}}{\text{apparent depth}}$ .
Shift (apparent rise) $= \text{real depth} - \text{apparent depth} = \text{real depth}\,(1 - 1/\mu)$ .
Viewed obliquely, the object appears raised even more.
Effects: a pool looks shallower; a stick dipped obliquely looks bent at the surface; print under a glass block looks raised; to a fish, a bird looks higher.

Table of real depth, refractive index, apparent depth and shift for three cases
Real depth	μ	Apparent depth	Shift
12 cm	4/3	9 cm	3 cm
9 cm	1.5	6 cm	3 cm
5 m	1.25	4 m	1 m

Formulas

Real and apparent depth
$\mu = \frac{\text{real depth}}{\text{apparent depth}}$
Apparent rise for real depth d
$\text{shift} = d\left(1 - \frac{1}{\mu}\right)$

Key terms

Apparent depth: The depth at which an object in a denser medium appears to be when viewed from the rarer medium.
Shift: The distance by which an object under a denser medium appears raised: real depth minus apparent depth.

5. Refraction through a triangular prism

A prism has two refracting faces inclined at the angle of the prism $A$ .
The ray bends towards the normal at the first face and away from it at the second; the emergent ray is turned towards the base.
Angle of deviation $\delta$ : angle between the incident ray produced and the emergent ray.
Relations: $r_{1} + r_{2} = A$ ; $\delta = i + e - A$ .
$\delta$ depends on (i) angle of incidence, (ii) angle of the prism, (iii) refractive index of the material, (iv) colour of light (violet deviates most). As $i$ increases, $\delta$ first decreases to a minimum, then increases.

Table of factors that change the angle of deviation of a prism
Change	Effect on deviation
Larger prism angle A	Larger
Larger refractive index	Larger
Violet light instead of red	Larger
Angle of incidence increased	First less, then more

Formulas

Angles inside a prism
$r_{1} + r_{2} = A$
Deviation
$\delta = i + e - A$

Key terms

Angle of deviation: The angle between the direction of the incident ray and the direction of the emergent ray.
Angle of the prism: The angle between the two refracting faces of a prism.

6. Critical angle and total internal reflection

Critical angle C: the angle of incidence in the denser medium for which the angle of refraction in the rarer medium is $90^\circ$ .
For a medium in air: $\mu = 1 / \sin C$ .
Total internal reflection (TIR): when $i > C$ , all the light is reflected back into the denser medium. Conditions: light travels from denser to rarer; $i > C$ .
Totally reflecting prisms (45°–45°–90° glass, $C \approx 42^\circ$ ): enter a short face normally → turned through 90° (periscope); enter the hypotenuse face normally → turned through 180° (binoculars, reflectors). An equilateral prism can also reflect totally at a face.
Advantages over a plane mirror: all light reflected (brighter), one image only, no silvering to spoil.

Table of critical angles of water, glass and diamond
Medium	C (about)
Water	49°
Glass	42°
Diamond	24°

Formulas

Critical angle
$\mu = \frac{1}{\sin C}$

Key terms

Critical angle: The angle of incidence in the denser medium for which the refracted ray grazes the surface (angle of refraction 90°).
Total internal reflection: Reflection of all the light back into a denser medium when it meets the boundary with a rarer medium at more than the critical angle.

Exam tips

Common mistakes

Measuring angles from the surface instead of the normal.
Saying the frequency of light changes in glass.
Writing apparent depth = real depth × μ.
Giving only one condition for total internal reflection.
Drawing the emergent ray of a glass block not parallel to the incident ray.

How marks are usually awarded

Calculations: formula, substitution, answer (and unit where there is one; μ has none).
Ray diagrams: arrows on rays, normals drawn dotted, angles labelled, emergent ray direction correct.
TIR: both conditions, each with its key word (denser to rarer; greater than the critical angle).

1. Refraction and its laws

Formulas

Key terms

2. Refractive index

Formulas

Key terms

3. Rectangular glass block and thick mirror

Formulas

Key terms

4. Real and apparent depth

Formulas

Key terms

5. Refraction through a triangular prism

Formulas

Key terms

6. Critical angle and total internal reflection

Formulas

Key terms

Exam tips

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