Lenses, Prisms and Dispersion: Quick Revision

A brief note on Lenses, Prisms and Dispersion. Read it before a test. Try each question at the end before you open its answer.

The thin lens

• Signs (New Cartesian): light goes left to right, distances are measured from the optical centre, and a real object has $u < 0$.
• Thin lens formula: $\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$.
• Magnification: $m = \dfrac{v}{u}$. A negative $m$ is an inverted image.
• Real is positive (A-Level, IB, AP): $\dfrac{1}{u} + \dfrac{1}{v} = \dfrac{1}{f}$ and $m = -\dfrac{v}{u}$.
• Lens maker's formula: $\dfrac{1}{f} = (n-1)\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$. A biconvex lens has $R_1 > 0$ and $R_2 < 0$.
• In a medium: replace $n$ by $\dfrac{n_g}{n_m}$. A concave lens always gives a virtual, upright, smaller image of a real object.

Power and combinations

• Power: $P = \dfrac{1}{f}$, with $f$ in metre. The unit is the dioptre.
• In contact: $P = P_1 + P_2$ and $m = m_1m_2$.
• A gap $d$: $P = P_1 + P_2 - dP_1P_2$. The pair is afocal when $d = f_1 + f_2$.
• Direct method: the image of the first lens is the object of the second, $u_2 = v_1 - d$.

The prism and dispersion

• Angles: $r_1 + r_2 = A$ and $\delta = i + e - A$.
• Minimum deviation: $i = e$ and $n = \dfrac{\sin\frac{A + \delta_m}{2}}{\sin\frac{A}{2}}$.
• Thin prism: $\delta = (n-1)A$.
• Dispersive power: $\omega = \dfrac{n_b - n_r}{n_y - 1}$. Violet bends most. A flat sheet does not disperse.
• Sky: scattering $\propto \dfrac{1}{\lambda^4}$, so the sky is blue and the sunset red. The primary bow is at $42^\circ$ (red outside), the secondary at $51^\circ$ (red inside).

A worked example

A thin convex lens has $f = 12\ \text{cm}$. An object is $20\ \text{cm}$ in front. Find the image.

1. Distance. $v = \dfrac{uf}{u + f} = \dfrac{(-20)(12)}{-20 + 12}\ \text{cm} = 30\ \text{cm}$.
2. Magnification. $m = \dfrac{30}{-20} = -1.5$. The image is real, inverted and larger.

Practise with new numbers

Check yourself

1. What is the power of a lens of focal length $-25\ \text{cm}$?

2. A thin prism of angle $6^\circ$ and index $1.5$ deviates a ray by how much?

3. Why does a prism split white light and a flat sheet of glass does not?

Sources

• NCERT Physics Part II (Class 12), Chapter 9.
• OpenStax University Physics Volume 3, Chapters 1 and 2.