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

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Differentiating the electrostatic potential with respect to spatial coordinates allows for the derivation of the electric field expression. The electric field is represented as a negative gradient of the potential, incorporating partial derivatives with respect to x, y, and z. When differentiating, dq(r') is treated as a constant since it is independent of the unprimed coordinates. The relationship between r and r' is clarified by recognizing that the variables in the Green function are assumed to be independent. Understanding this distinction is crucial for correctly applying the differentiation process.
Phymath
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By differentiating the electrostatic potential

<br /> \Phi(\vec{r}) = \int\int_{\Omega} \frac{k_e dq(\vec{r&#039;})}{|\vec{r}-\vec{r&#039;}|}

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