(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose the electric field in some region is found to be [itex]\vec{E} = kr^{3} \hat{r}[/itex], in spherical coordinates (k is some constant).

(a) Find the charge density ρ.

(b) Find the total charge contained in a sphere of radius R, centered at the origin. (Do it two different ways.)

2. Relevant equations

3. The attempt at a solution

(a) Use the formula for the divergence of a vector in spherical basis to get [itex]\nabla . \vec{E} = 5kr^{2}[/itex] so that [itex]\rho = 5k\epsilon_{0}r^{2}[/itex].

(b) Q is the volume integral of [itex]\rho[/itex] over the volume of the sphere. So, we integrate over d[itex]\phi[/itex], integrate sinθ over dθ, integrate the [itex]\rho[/itex] times [itex]r^{2}[/itex] over dr and multiply the three results. The process gives us 4[itex]\pi \epsilon_{0} R^{5}[/itex].

Q can also be found using the integral form of Gauss's law, where the surface integral of the electric field is taken with the infinitesimal area, which is R^{2}sinθ dθ d∅r, whereris the unit vector in the radial direction. We take the constants out of the integral and integrate 1 over phi and sinθ over theta to obtain the same Q as above.

Please could you check if the process and the answers are correct?

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# Homework Help: Gauss's law in action

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