- #1
Jaccobtw
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- 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
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