A glass cube has side length a and its refractive index is $μ = 3 / 2$ . A ray of light (AB) is incident normally on one of its face and after passing through the cube it forms a spots S on screen $Σ$ . The cube begins to rotate with angular speed ' $ω$ ' about its central axis. Immediately after cube begins to rotate, the speed of spot `S` on screen is $\frac{aω}{K}$ . The value of K is _______

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answer is 3.

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Detailed Solution

Lateral shift when i is small is

$d = a (1 - \frac{1}{μ}) i$

The spot at S is at a distance y from O (the point where incident ray will hit screen if there is no shift)

$y = \frac{d}{\cos i}$

When $i \to 0; y \to d$

$y = a (1 - \frac{1}{μ}) i$

$\begin{array}{l} \frac{dy}{dt} = a (1 - \frac{1}{μ}) \frac{di}{dt} \\ \Rightarrow v = a (1 - \frac{1}{μ}) . ω = \frac{aω}{3} \\ ∴ k = 3 \end{array}$

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