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(A) For a medium with a variable refractive index $\mu(y)$,the apparent depth $h_{app}$ is given by the integral of the ratio of the infinitesimal dep

Vedclass · Accepted Answer

$\frac{\pi}{4} \ m$

Vedclass · Answer

(A) For a medium with a variable refractive index $\mu(y)$,the apparent depth $h_{app}$ is given by the integral of the ratio of the infinitesimal depth element $dy$ to the refractive index at that depth: $h_{app} = \int_{0}^{h} \frac{dy}{\mu(y)}$ Given $\mu(y) = y^2 + 1$ and the actual depth $h = 1 \ m$,we substitute these into the integral: $h_{app} = \int_{0}^{1} \frac{dy}{y^2 + 1}$ Using the standard integral formula $\int \frac{dx}{x^2 + 1} = 	an^{-1}(x) + C$,we get: $h_{app} = [	an^{-1}(y)]_{0}^{1}$ $h_{app} = 	an^{-1}(1) - 	an^{-1}(0)$ $h_{app} = \frac{\pi}{4} - 0 = \frac{\pi}{4} \ m$ Thus,the apparent depth is $\frac{\pi}{4} \ m$.

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