How to Differentiate the Electrostatic Potential to Derive the Electric Field?

[Phymath]

By differentiating the electrostatic potential

$$\Phi(\vec{r}) = \int\int_{\Omega} \frac{k_e dq(\vec{r'})}{|\vec{r}-\vec{r'}|}$$

with respect to x, y, and z, and asumming that $$\Omega$$ is independent of x,y, and z show the electric field, can be written as

$$\vec{E}=\frac{-\partial{\Phi}}{\partial{x}}\hat{\vect{e_x}}-\frac{\partial{\Phi}}{\partial{y}}\hat{\vect{e_y}}-\frac{\partial{\Phi}}{\partial{z}}\hat{\vect{e_z}}$$

the problem is how to do I do the diriv of the the dq(r') function? no idea...probley chain rule any hints also help

 

[StatusX]

You're differentiating with respect to the unprimed coordinates so dq(r) is treated as a constant.

 

[dextercioby]

There's no connection between "r" and "r' ".You can see that by taking a look at the derivation of that formula...Namely the variables of the Green function are naturally assumed to be independent...


Daniel.