Question

Option 1: Write the 'lens maker' formula for a thin lens. The refractive index of the material of a thin lens is \( n \) and its focal length is \( f \). The lens is placed in a medium of refractive index \( n \). Find the focal length of the lens in the medium.
OR
Option 2: What is the difference between unpolarized and polarized light? Two polaroids A and B are placed such that light polarized by A cannot pass through B. Can some light be obtained by using a third polaroid?

Hint:

Use \( \frac{1}{f}=\left(\frac{n}{n_m}-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right) \); when the medium index equals the lens index the power is zero. For the polaroids, apply Malus' law in two stages through the middle polaroid.

Answer 1

OPTION 1 (Lens maker formula in a medium)

Step 1: Lens maker formula. For a thin lens the formula is \[ \frac{1}{f} = \left(\frac{n_{lens}}{n_{med}} - 1\right)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \] where \( n_{lens} \) is the refractive index of the lens material, \( n_{med} \) that of the surrounding medium, and \( R_1, R_2 \) the radii of curvature of the two surfaces.

Step 2: Focal length in air. In air \( n_{med} = 1 \), so \[ \frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \quad\text{...(i)} \]
Step 3: Focal length in a medium of index } n_m. If the lens is now placed in a medium of refractive index \( n_m \), its focal length \( f_m \) is \[ \frac{1}{f_m} = \left(\frac{n}{n_m} - 1\right)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \quad\text{...(ii)} \]
Step 4: Divide (i) by (ii). The bracket \( \left(\frac{1}{R_1}-\frac{1}{R_2}\right) \) cancels: \[ \frac{f_m}{f} = \frac{(n-1)}{\left(\dfrac{n}{n_m}-1\right)} = \frac{(n-1)\,n_m}{(n - n_m)} \] so the general result is \[ \boxed{\,f_m = \dfrac{(n-1)\,n_m}{(n - n_m)}\;f\,} \]
Step 5: Put the medium index equal to the lens index. Here the medium has the same refractive index as the lens material, i.e. \( n_m = n \). Substituting, the denominator \( (n - n_m) = 0 \), therefore \[ \boxed{\,f_m \to \infty\,} \] The lens loses all its converging or diverging power and behaves like a plane glass plate. Physically, when the surrounding medium has the same optical density as the lens, there is no bending of light at either surface, so the focal length becomes infinite.

OPTION 2 (Polarized light and a third polaroid)

Step 1: Unpolarized vs polarized light. In unpolarized light the electric field vibrations occur equally in all directions in the plane perpendicular to the direction of propagation. In polarized (plane polarized) light the vibrations are confined to a single direction perpendicular to propagation.

Step 2: The crossed pair A and B. Polaroid A produces plane polarized light. B is oriented so that light from A cannot pass, meaning A and B are 'crossed' (their transmission axes are at \( 90^\circ \)). By Malus' law the intensity through B is \( I = I_0\cos^2 90^\circ = 0 \), so no light emerges.

Step 3: Insert a third polaroid C between A and B. Let its axis make angle \( \theta \) with the axis of A. Intensity after A \( = I_0 \). After C: \( I_C = I_0\cos^2\theta \). The angle between C and B is \( (90^\circ - \theta) \), so after B: \[ I_B = I_0\cos^2\theta\,\cos^2(90^\circ-\theta) = I_0\cos^2\theta\,\sin^2\theta = \frac{I_0}{4}\sin^2 2\theta \]
Step 4: Conclusion. For any \( \theta \) with \( 0^\circ < \theta < 90^\circ \), \( I_B \) is not zero. Hence yes, some light does come out of B. It is maximum ( \( I_0/4 \) ) when \( \theta = 45^\circ \). \[ \boxed{\,\text{A third polaroid between A and B lets light emerge; } I_{max}=I_0/4 \text{ at } 45^\circ\,} \]