Question

Consider the earlier problem where we had a point source radiating energy at the rate of 10 Joules per second, or 10 Watts. What can you say about the intensities $I_{1}and{I}_2$ at two points on spheres of radii and $r_{1}and{r}_2$ respectively, centered at the source?

• A. $I_1=I_2$
• B. $I_1{r}_1=I_2{r}_2$
• C. $I_1{r}_1^2=I_2{r}_2^2$
• D. $I_1{r}_2^2=I_2{r}_1^2$

Answer 1

The correct option is C

$I_1{r}_1^2=I_2{r}_2^2$

The diagram shows a point source radiating energy at 10 Watts, or 10 Joules every second. This energy spreads out equally in all directions (isotropically), and travels at the constant speed of light. This can be imagined as a spherical shell of energy centered around the source, expanding at the wave speed of(speed of light), just like circular waves travelling outwards when you drop a pebble on a calm lake.

10 J of energy radiated spreads out spherically every second. When we look at the inner sphere, 10 J spreads out evenly over an area of $4\pi r_1^2$ leading to intensity $I_1=\frac{10}{4\pi r_1^2}W/m^2.$ at each point on the inner shell.

Similarly, points on the outer shell will have intensity $I_2=\frac{10}{4\pi r_2^2}W/m^2.$

Dividing, we see,

$\frac{I_1}{I_2}=\frac{r_2^2}{r_1^2}{I}_1{r}_1^2=I_2{r}_2^2=constant.$

This is intuitive, right? When we move away from a light source, the brightness (which is a measure of the intensity) decreases, so that the product $I{r}^2$ stays constant.