- #1
Opus_723
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Homework Statement
First, the problem, quoted verbatim:
"Neutrons are created (by a nuclear reaction) inside a hollow sphere of radius R. The newly created neutrons are uniformly distributed over the spherical volume. Assuming that all directions are equally probable (isotropy), what is the average distance a neutron will travel before striking the surface of the sphere? Assume straight line motion, no collisions."
The problem then has several "steps" to help you along. I'm stuck on the first one, setting up the integral,
[itex]\bar{r}=3R/2\int^{1}_{0}\int^{\pi}_{0}\sqrt{1-k^{2}sin^{2}\theta}*k^{2}sin\theta*dk*d\theta[/itex]
The Attempt at a Solution
I've mostly got it, except that my expression under the square root is wrong, so I think I'm having a conceptual issue with how to calculate the average distance. I found the distance from a point within the sphere at distance a from the orgin on the z-axis to a given point on the sphere using the law of cosines:
d=[itex]\sqrt{R^{2}+a^{2}-2Racos\theta}[/itex]
Using this distance, I integrated over theta, phi, and a, then divided by the volume of the sphere. After substituting Rk = a, I got:
[itex]\bar{r}=3R/2\int^{1}_{0}\int^{\pi}_{0}\sqrt{1+k^2-2kcos\theta}*k^{2}sin\theta*dk*d\theta[/itex]
So clearly I'm making a mistake with the distance. Could someone set me straight?