Derive an expression for the radial charge distribution of an E field

[Jaccobtw]

Homework Statement

Consider a spherical distribution of charge that creates a uniform, radial electric field described by: $\vec{E}=E_o\hat{r}$

Use the differential form of Gauss's Law to derive an expression for the radial charge distribution, $\rho$, that will create this field. You will need the divergence in spherical coordinates: $\nabla \cdot \vec{V}=\frac{1}{r^2}\frac{\partial}{\partial{r}}(r^2 V_r)+\frac{1}{r\sin[{\theta}]}\frac{\partial}{\partial{\theta}}(\sin[{\theta}]V_{\theta})+\frac{1}{r\sin[{\theta}]}\frac{\partial}{\partial{\phi}}(V_{\phi})$
Enter your mathematical expression for ρ(r) in terms of $\epsilon_o, E_o$, and $r$

Relevant Equations

$$\nabla \cdot \vec{V}=\frac{1}{r^2}\frac{\partial}{\partial{r}}(r^2 V_r)+\frac{1}{r\sin[{\theta}]}\frac{\partial}{\partial{\theta}}(\sin[{\theta}]V_{\theta})+\frac{1}{r\sin[{\theta}]}\frac{\partial}{\partial{\phi}}(V_{\phi}) $$

I know we're supposed to attempt a solution but I'm honestly super confused here. I think the second an third terms of the del equation can be cancelled out because there is only an E field in the r hat direction, so no e field in the theta and phi directions. That leaves us with $\nabla \cdot \vec{E}=\frac{1}{r^2}\frac{\partial}{\partial{r}}(r^2 E_r)$. The question says to write an expression for $\rho (r)$. I know gauss' law has the differential form of itself equaling $\frac{\rho}{\epsilon_o}$. would you isolate $\rho$ to get an expression for $\rho (r)$? Thanks for your help

 

[TSny]

I think the second an third terms of the del equation can be cancelled out because there is only an E field in the r hat direction, so no e field in the theta and phi directions.

Yes.

That leaves us with $\nabla \cdot \vec{E}=\frac{1}{r^2}\frac{\partial}{\partial{r}}(r^2 V_r)$.

The symbol $V_r$ on the right side would be better written as $E_r$.

The question says to write an expression for $\rho (r)$. I know gauss' law has the differential form of itself equaling $\frac{\rho}{\epsilon_o}$. would you isolate $\rho$ to get an expression for $\rho (r)$?

Yes. See what you get.

 

[Jaccobtw]

Yes.The symbol $V_r$ on the right side would be better written as $E_r$.Yes. See what you get.

Thank you. I got $\frac{2 E_o \epsilon_o}{r}$

 

[TSny]

Thank you. I got $\frac{2 E_o \epsilon_o}{r}$

Looks good.