# How long will the light exist inside the sphere?

Ok, so let's say we have a very thick, but hollow, metal sphere. The inside of the sphere has a radius of 100 meters. The sphere also has a door that can be opened and closed and when closed makes an extremely strong seal. The door is approximately 10x10 meters. When closed, the interior wall of the sphere is a continuous and perfect mirror. The sphere is also floating in space next to earth in direct view of the sun.

So let's say I open the door for a minute, then close the door. How long will the light exist and illuminate the inside of the sphere, approximately? If I place the sphere next to planet Mercury and then close the door for a minute, will the inside of the sphere stay illuminated longer? Where does the light go?

• In addition to all your "perfections", you also need to be able to close the door fast enough for the light don't have time to go away. You have to make it fast ! (indeed the duration of opening is not important, because all leaks away as long as the door is not closed). Beside, yes, closer to the Sun, the initial flux is larger. – Fabrice NEYRET Oct 30 '15 at 16:57

## 1 Answer

Your assumption of a perfect mirror is something that will make this difficult to answer (also assuming you can close the door fast enough to keep some amount of the light in before it escapes). If the mirror is perfect and no light escapes, then it should contain the light, bouncing around forever. In reality, a small portion of the light is absorbed upon each reflection. If you have Reflectivity, $R = 0.999$, for instance (which would be considered very good), still 0.1% of the light is absorbed by the mirror and turned into heat. So, lets start with this assumption. The sphere has a diameter of approximately $r = 1 \text{ m}$, with the reflectivity coefficient, $R$ from above. The fraction of light that remains, $L$, after $N$ bounces of light on the mirror is given by

$$L = R^N$$

The amount of time that it takes to do $N$ bounces is $t = Nr/c$, where $c$ is the speed of light. So, the fractional amount of light left after a time $t$ is given by

$$L(t) = R^{tc/r}$$

Since $R$ is less than one, this will be an exponential decay. So lets plug these numbers in.

For $t= 1$ $\mu$s, $L = 0.74$.

For $t= 10$ $\mu$s, $L = 0.05$.

For $t= 100$ $\mu$s, $L = 9.2\cdot 10^{-14}$.

So the decay is extremely fast, as you might expect from exponential decay. Feel free to plug in your own numbers, but for even the slightest imperfection in the mirror, the decay will build up quickly due to the exponential nature of things.

Since this is fractional, it really won't depend much on where you place the sphere, the total amount of light is different, but the fractional decay will be the same.

• With a perfect mirror, the only mechanism for decay would be the light absorbed by the observer. I suspect that this would also be an exponential decay. I also suspect absorbing thousands of joules of solar energy would be very uncomfortable. – Robert Stiffler Oct 30 '15 at 20:22