Rays of light are incident symmetrically on a sphere of radius $$R$$ as shown (below middle). Determine the refractive index of the sphere material if they meet at P (You need to write the correct expression for refractive index). Can the rays be made to meet at C, the center of the sphere. If not, how should the rays be incident to meet at C.
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PLACEHOLDER ANSWER - the answer is an expression and a description. Taking P as the point where the axis meets the far surface, each ray (at height $$\frac{R}{2}$$, as in the figure) has angle of incidence $$i$$ with $$\sin i = \frac{1}{2}$$, so $$i = 30^\circ$$. The refracted ray is a chord ending at P, and the angle it makes with the radius at the point of entry is half the angle at the centre, so $$r = \frac{i}{2}$$ and $$\mu = \frac{\sin i}{\sin(i/2)} = 2\cos\frac{i}{2} = 2\cos 15^\circ \approx 1.93$$. Rays parallel to the axis cannot be made to meet at C, since a refracted ray passes through the centre only if it travels along a radius; to meet at C, the rays must strike the sphere normally, i.e. be directed along radii towards C, and they then pass undeviated.