A point light source is moving with constant velocity \(v\) inside a transparent thin spherical shell of radius \(R\), which is filled with a transparent liquid. If at \(t=0\) , light source is at the centre of the sphere, then at what time a thin dark ring will be visible for an observer outside the sphere. the refractive index of liquid with respect to that of shell is \(\sqrt{2}\).

The time is of form \( \frac{R}{kv} \), where k is a real number . Find k.

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