Question

A pointsource of light is$\mathbf{80}\mathbf{.0}\mathbf{\text{cm}}$below the surface of a body of water. Find the diameter of the circle at the surface through which light emerges from the water.

Short answer

$DiameterofCircle=1.82m$

Step-by-step solution

Given

Depth of the source with respect to surface$=80cm=0.8m$

Understanding the concept

We can draw a ray diagram for the given problem. From the ray diagram, we can get the formula for the diameter of the circle. The value of critical angle can be found by using Snell’s law of refraction. Inserting it in that formula will give the diameter of the circle at the surface through which light emerges from the water.

Calculate the diameter of the circle at the surface through which light emerges from the water

Ray Diagram for the situation:

From the above diagram, we can say that

$\begin{array}{c}tan(90-{\theta }_C)=\frac{Depth}{RadiusofCircle}\\ DiameterofCircle=2\times radiusofcircle\\ RadiusofCircle=\frac{DiameterofCircle}{2}\\ tan(90-{\theta }_C)=\frac{Depth}{\left(\frac{DiameterofCircle}{2}\right)}\end{array}$

$\begin{array}{c}cot{\theta }_C=\frac{Depth}{\left(\frac{DiameterofCircle}{2}\right)}\\ tan{\theta }_C=\frac{\left(\frac{DiameterofCircle}{2}\right)}{Depth}\\ tan{\theta }_C=\frac{DiameterofCircle}{2\times Depth}\\ DiameterofCircle=2\times Depth\times tan{\theta }_C\end{array}$

We have

$n_w=1.33$And$n_a=1$

From the figure, we can say that for air, the refractive angle is.

So according to Snell’s Law,

$\begin{array}{c}{n}_{1}sin{\theta }_1=n_{2}sin{\theta }_2\\ n_{w}sin{\theta }_C=n_{a}sin{\theta }_a\\ sin{\theta }_C=\frac{n_a}{n_w}sin{\theta }_a\\ sin{\theta }_C=\frac{1}{1.33}\times sin90°\end{array}$

$\begin{array}{c}sin{\theta }_C=0.75\\ {\theta }_C=48.7°\end{array}$

We have

$\begin{array}{l}DiameterofCircle=2\times DepthoftheSource\times tan{\theta }_C\\ DiameterofCircle=2\times 0.8\times tan(48.7)\\ DiameterofCircle=1.82m\end{array}$