- #1

- 128

- 1

## Homework Statement

In some region of space, the electric field is [itex] \vec{E} =k r^2 \hat{r} [/itex], in spherical coordinates, where k is a constant.

(a) Use Gauss' law (differential form) to find the charge density [itex] \rho (\vec{r}) [/itex].

(b) Use Gauss' law (integral form) to find the total charge containted in a sphere of radius R, centered at r=0

(c) Find the total charge contained in a sphere of radius R, centered at r=0 by direct integration of your result for [itex] \rho (\vec{r}) [/itex] in (a). Does the answer agree with that in (b)?

## The Attempt at a Solution

NOTE: IM USING [itex] \Delta [/itex] as the del operator

Im pretty sure my problems only arise on part (c), but if any error is noticed in the other parts please tell me. Thank you

(a)using [tex] \vec{ \Delta} \bullet \vec{E} = \frac{\rho}{\epsilon_0} [/tex]

rearranging it to solve for [itex] \rho [/itex] i get [tex] \rho = \epsilon( \vec{\Delta} \bullet \vec{E}) [/tex] where [tex] \epsilon( \vec{\Delta} \bullet \vec{E}) = \frac{\epsilon_0}{r^2} \frac{d(r^2 k r^2)}{dr}[/tex] which reduces to [tex] \frac{r}{\pi} [/tex]

(b) using [tex] \oint \Delta \bullet d\vec{a} = \frac{Q_e}{\epsilon_0} [/tex] and rearranging to solve for Q_e i get [tex] Q_e=\epsilon_0 \oint \vec{E} \bullet d\vec{a} [/tex], which since the sphere is symmetric about the origin i can do [tex] \epsilon_0 \oint |\vec{E}| d\vec{a} [/tex], which equals [tex] \epsilon_0 k r^2 4\pi R^2 [/tex] which reduces to [tex] R^4 [/tex] (note: i replaced [itex] r^2 [/itex] with [itex] R^2 [/itex])

(c) knowing [itex] \rho = \frac{dq}{dV} [/itex] and reorganizing to solve for dq i get [itex] \int dq = \int \rho(\vec{r}) dV [/itex] at this point I am a little confused on how to take the integral with respect to dV in spherical coordinates. I am pretty sure i have to add an [itex] r^2 [/itex] in the integrand but I am not sure.