Particle density in spherical geometry

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Homework Statement

Neutrons are emitted uniformly from the inner surface of a thin spherical shell of radius R at a velocity V. They are emitted normal to the inner surface and fly radially across the volume of the sphere to be absorbed at diametrically opposed points. The neutrons are non interacting and do not collide. Express the neutron density at radius r as a function of the emitted neutron current and neutron velocity.

Homework Equations

I don't really have any, other than some simple continuity equations and equations for the area and volume of a sphere.

The Attempt at a Solution

I can think of an ad hoc way to do this, by taking the ratios of the surface areas, and expressing the density as some function of this ratio, but it doesn't seem correct.

 

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I've rationalised that for linear geometry ρ=I/AV where I is the current, A the surface area and V is the velocity. Applying this relationship to this scenario simply yields ρ=I/V4∏r^2

This doesn't seem quite right though.