Refraction Through Prisms and Lenses: Class 11 Physics Formulas, Derivations and Solved Numericals

Geometrical optics in NEB Class 11 Physics runs from reflection at curved mirrors through refraction at plane surfaces, prisms, lenses and dispersion. The prism and lens chapters carry the numericals: finding the refractive index of a prism from its minimum deviation, locating an image through a lens, and finding a focal length from the lens maker's formula. The derivations behind these formulas are asked as often as the numericals themselves.

This guide gives the formulas in exam order with the derivation steps the examiner marks and one worked example each. The complete chapter notes and solved board questions are on the Refraction at Plane Surface, Refraction Through Prisms and Lenses chapter pages, with Dispersion following.

Refraction at a plane surface

Refraction is the bending of light at the boundary between two media because its speed changes. The laws of refraction: the incident ray, refracted ray and normal lie in one plane, and for a given pair of media

$$\frac{\sin i}{\sin r} = {}_a\mu_b$$

which is Snell's law. The absolute refractive index of a medium is $\mu = c/v$, always greater than 1. By the principle of reversibility, ${}_w\mu_a = 1/{}_a\mu_w$.

Lateral shift

A ray through a parallel-sided slab of thickness $t$ emerges parallel to its original direction but displaced sideways by

$$d = \frac{t,\sin(i - r)}{\cos r}$$

At grazing incidence ($i = 90^\circ$) the shift equals the thickness, $d = t$.

Real and apparent depth

An object in a denser medium viewed from above appears raised. For near-normal viewing,

$$\mu = \frac{\text{real depth}}{\text{apparent depth}}$$

Worked example 1: apparent depth of a pool

A swimming pool is 3 m deep. How deep does it appear to a person looking straight down, if the refractive index of water is 4/3?

$$\text{apparent depth} = \frac{3}{4/3} = 2.25\ \text{m}$$

The bottom appears raised by 0.75 m, which is why pools look shallower than they are.

Critical angle and total internal reflection

When light travels from a denser to a rarer medium, the angle of refraction exceeds the angle of incidence. The angle of incidence for which the refracted ray grazes the surface ($r = 90^\circ$) is the critical angle $C$:

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

For incidence beyond $C$ the light is totally internally reflected. Optical fibres, the sparkle of a diamond and mirages are the standard applications.

Worked example 2: critical angle of glass

Find the critical angle for glass of refractive index 1.5.

$$\sin C = \frac{1}{1.5} = 0.667 \quad\Rightarrow\quad C = 41.8^\circ$$

Refraction through a prism

A prism is a transparent medium bounded by two plane faces inclined at the refracting angle $A$. A ray entering at angle of incidence $i$ refracts at $r_1$, crosses the prism, meets the second face at $r_2$ and emerges at angle $e$. The angle between the incident and emergent directions is the deviation $\delta$.

Derivation of $i + e = A + \delta$

The deviation at the first face is $(i - r_1)$ and at the second face $(e - r_2)$, so

$$\delta = (i + e) - (r_1 + r_2)$$

In the quadrilateral formed by the two faces and the two normals, the normals meet the faces at right angles, so the angle between the normals inside the prism is $180^\circ - A$. In the triangle formed by the ray and the two normals, $r_1 + r_2 + (180^\circ - A) = 180^\circ$, giving

$$A = r_1 + r_2$$

Substituting,

$$i + e = A + \delta$$

Minimum deviation and the prism formula

As $i$ increases, $\delta$ falls to a minimum $\delta_m$ and then rises. At minimum deviation the ray passes symmetrically, so $i = e$ and $r_1 = r_2 = r$. Then $A = 2r$ gives $r = A/2$, and $2i = A + \delta_m$ gives $i = (A + \delta_m)/2$. Snell's law at the first face, $\mu = \sin i / \sin r$, becomes

$$\mu = \frac{\sin\left(\dfrac{A + \delta_m}{2}\right)}{\sin\left(\dfrac{A}{2}\right)}$$

This is the formula used in the laboratory to measure the refractive index of a prism.

Worked example 3: refractive index from minimum deviation

An equilateral prism ($A = 60^\circ$) produces a minimum deviation of $30^\circ$. Find its refractive index.

$$\mu = \frac{\sin 45^\circ}{\sin 30^\circ} = \frac{0.7071}{0.5} = 1.414$$

Small-angle prism

When $A$ is small every angle is small and $\sin\theta \approx \theta$, so $i = \mu r_1$ and $e = \mu r_2$. Then $\delta = \mu(r_1 + r_2) - (r_1 + r_2) = (\mu - 1)A$:

$$\delta = (\mu - 1)A$$

The deviation of a thin prism does not depend on the angle of incidence. This result is the basis of the dispersion chapter, where $\mu$ differs for each colour.

Worked example 4: deviation by a thin prism

A thin prism of angle $5^\circ$ is made of glass with $\mu = 1.5$. Find the deviation it produces.

$$\delta = (1.5 - 1) \times 5^\circ = 2.5^\circ$$

Lenses

A lens is a transparent medium bounded by two surfaces, at least one of them curved. A convex lens converges a parallel beam to a real focus; a concave lens diverges it from a virtual focus. The focal length $f$ is the distance from the optical centre to the principal focus.

All formulas below use the New Cartesian sign convention: distances are measured from the optical centre, positive in the direction of the incident light and negative against it. A convex lens has positive $f$, a concave lens negative $f$, and a real object on the left has negative $u$.

Lens formula and magnification

$$\frac{1}{f} = \frac{1}{v} - \frac{1}{u}, \qquad m = \frac{v}{u}$$

A negative magnification means an inverted (real) image; a positive one means erect (virtual).

Worked example 5: image through a convex lens

An object is placed 30 cm in front of a convex lens of focal length 20 cm. Find the position, nature and size of the image.

$u = -30$ cm, $f = +20$ cm.

$$\frac{1}{v} = \frac{1}{f} + \frac{1}{u} = \frac{1}{20} - \frac{1}{30} = \frac{3 - 2}{60} = \frac{1}{60}$$

$$v = +60\ \text{cm}, \qquad m = \frac{v}{u} = \frac{60}{-30} = -2$$

The image is 60 cm on the far side of the lens, real, inverted and twice the size of the object.

Lens maker's formula

For a thin lens of refractive index $\mu$ in air with surface radii $R_1$ and $R_2$ (signs by the convention),

$$\frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$

The derivation applies the refraction formula for a single spherical surface twice, once at each face, and treats the image from the first surface as the object for the second. In an answer, state the single-surface relation, write it for both surfaces, add the two, and note that the object and image distances cancel to leave the focal length.

Worked example 6: focal length of a biconvex lens

A biconvex lens has both surfaces of radius 20 cm and is made of glass with $\mu = 1.5$. Find its focal length.

With the convention, $R_1 = +20$ cm and $R_2 = -20$ cm.

$$\frac{1}{f} = (1.5 - 1)\left(\frac{1}{20} + \frac{1}{20}\right) = 0.5 \times \frac{2}{20} = \frac{1}{20}$$

$$f = 20\ \text{cm}$$

Power and combination

The power of a lens is $P = 1/f$ with $f$ in metres, in dioptres. Two thin lenses in contact have combined focal length $1/F = 1/f_1 + 1/f_2$ and powers that add.

Visual angle and angular magnification

The apparent size of an object depends on the angle it subtends at the eye, the visual angle. Angular magnification of an optical instrument is the ratio of the visual angle with the instrument to the visual angle without it, which is why a magnifying glass works: it lets the object be brought closer than the near point while still forming an image the eye can focus.

Formula sheet

How the optics chapters are examined

The derivations of $i + e = A + \delta$, the minimum deviation formula and the lens maker's formula are the long-answer regulars. Numericals come from minimum deviation, the lens formula and the lens maker's formula, occasionally from apparent depth or the critical angle. Draw the ray diagram for every answer, label the angles, and state the sign convention before using the lens formula. The solved board questions on each chapter page show the expected working, and the Class 11 Physics important questions view ranks the chapters by how often they appear. The preceding chapter, Reflection at Curved Mirrors, uses the same sign convention and is worth revising alongside.