How Is Fluence Rate Calculated at the Center of a Radiating Sphere?

[cbritta]

Homework Statement

The point P is located at the origin of a spherical-surface radiation source. The diameter of the sphere is 30 cm, and the source intensity is uniformly 10⁵ particles cm^-2 sec^-1 over the entire spherical surface. (a) Calculate the fluence rate(particles cm^-2 sec^-1) at P. (b) Calculate the average fluence rate over the entire sphere.

Homework Equations

Fluence Rate of an isotropic point source:Φ= S/4πr²

The Attempt at a Solution

For part a, I attempted to treat the entire spherical surface as a collection of point charges and integrate across this surface. I set up an integral from 0 to (4πr)^2, integrating the above equation for fluence rate of a point source Φ= S/4πr². The radius is a constant 30 cm and can be pulled out along with the rest of the fluence equation. After integrating I am getting S alone as the answer. The units are correct for the answer I am just not sure if I am going about this the right way. Furthermore, I have no idea about how to tackle part b without knowledge of part a.

 

[mark726]

For part a, your approach seems correct. You can think of the entire spherical surface as a collection of point sources, with each point source having the same intensity of 105 particles cm-2 sec-1. So, the fluence rate at P would be the same as that of a point source with intensity S=105 particles cm-2 sec-1 and radius r=30 cm.

For part b, you can use the definition of average fluence rate, which is the total fluence rate divided by the surface area. In this case, the total fluence rate is equal to the fluence rate at P (calculated in part a) multiplied by the surface area of the sphere. The surface area of a sphere is given by 4πr^2, where r is the radius of the sphere. So, the average fluence rate would be:

Φ_avg = (Φ at P) * (surface area of sphere) = (105 particles cm-2 sec-1) * (4π * (30 cm)^2) = 378,450 particles cm-2 sec-1

Hope this helps!