Problem no.14 "Illumination"

The following assumptions were made:

The problem is easy to solve, if the distance between the bulbs and the ball is very large . That means that the radius of the ball is small compared to the distance. In this case the intensity on both sides must be equal.
The intensity is proportional to the power of the source divided by the distance squared. If the intensity on both sides is equal, we have P1 divided by a1 squared is equal to P2 divided by a2 squared, where P1 and a1 are the power and the distance on one side and P2 and a2 on the other side. So the ratio between a1 and a2 is squareroot of P2 divided by P1. With P1 = 40 W and P2 = 100 W the ratio between the radii is approximately 0.63246.


If the ratio between the distance and the radius is small the problem is more complicate. The intensity varies at different positions on the ball. We think the two sides of the ball seem equally lit, if the power on both sides is equal.
On each imaginary sphere with the radius r and the centre in the light source, the energy that is radiated onto the sphere per second is equal to the energy emitted per second from the source.
A further step is to examine which part of the light radiated onto the sphere shines onto the ball as follows:

Calculation Calculation

In this formula Pr is the part of the light that shines onto the ball, Ps is the power of the source, A is the area of the big sphere that takes up the table tennis ball and r is the radius of the sphere.

Sketh to explain characters in the formulas
Sketch to explain characters in the formulas

For this last equation "Mathematica" provided us with an exact solution.

So we plotted the ratio aleft / aright versus aleft.
We can see, that if the distance between the bulb and the ball (aleft) goes to infinity the ratio is 0,63246. This is also the solution which we already knew from the assumption of large distances between bulbs and ball. So the result should be correct.

plot: a_left/a_right versus a a_left

If we forget our assumption that the filament of the bulb is a point source, we can say that one infinitesimal small part of the power is:
However not even for simple shapes like lines or arcs this equation is solvable. It is only possible in a numerical way.