A plane wavefront of width \(x\), is incident on an air-water interface and the corresponding refracted wavefront has a width y as shown in figure. The refreactive index of air with respect to water in terms of distances w and z is
Show Hint
In the same time t the wave travels z in air and w in water. So v_water / c = w / z.
Step 1: Understanding the Concept:
The figure shows a plane wavefront of width \(x\) meeting the air-water surface between points A and B. By Huygens' construction, the point of the wavefront that touches A first sends a wavelet into the water, while the other end of the wavefront is still travelling in air to reach B.
Step 2: Key Formula or Approach:
The refractive index of air with respect to water is the ratio of the speed of light in water to the speed in air:
\[ {}_w n_a = \frac{v_{\text{water}}}{v_{\text{air}}} \]
Step 3: Detailed Explanation:
Let \(t\) be the time in which the far end of the wavefront travels the distance \(z\) in air to reach B. Then
\[ z = v_{\text{air}}\,t \]
In the same time \(t\), the wavelet from A travels the distance \(w\) in water:
\[ w = v_{\text{water}}\,t \]
Divide:
\[ \frac{w}{z} = \frac{v_{\text{water}}}{v_{\text{air}}} \]
This is the refractive index of air with respect to water, which is less than 1 because light is slower in water. The inverse ratio \(z/w\) in option (B) is the refractive index of water with respect to air. Options (C) and (D) with square roots have no basis in this construction.
Final Answer:
The refractive index of air with respect to water is \(\dfrac wz\), option (A).
\[ \boxed{\frac{w}{z} \text{ (A)}} \]
