Question

The speed of light relative to still water is $\mathbf{2}\mathbf{.}\mathbf{25}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\mathit{m}\mathbf{/}\mathit{s}$ .If the water is moving past us, the speed of light we measure depends on the speed of the water. Do these facts violate Einstein’s second postulate? Explain.

Short answer

The speed of light in a medium with a refractive index u = c/n is thus, the speed of light we measure depends on the speed of the water

Step-by-step solution

Concept of the Einstein’s second postulate

Einstein's second postulate states that the speed of light in vacuum is the constant c in all inertial frames of reference and is independent of the motion of the source. However, in a medium with a different density, like water, light is known to propagate with a less speed u, characterized by the refractive index n of that medium: u = c/n , where n > 1 (for vacuum n =1). Also, the speed of water we measure should depend on the speed of the moving water. To see why, let us call our frame of reference S (in which we are at rest) and the frame of reference of the moving water S'. The frame S' is moving past S with a relative speed, say v. The speed of light relative to S' is $\mathrm{u}\text{'}=2.25\times {10}^8\mathrm{m}/\mathrm{s}$Suppose light travels in the same direction as the moving water (frame S'), then the speed of water as measured by us (in frame S) is given by

$u=\frac{u\text{'}+v}{1+u\text{'}v/c^2}$

As we can see, the speed of light, u, we measure does depend on the speed of water v.