Question

Formula for electric field

Answer 1

For the point source with charge q the electric field $\vec{E}$ is $\vec{E} = \dfrac{1}{4\pi \varepsilon_0}\cdot \dfrac{q}{r^3}\cdot\vec{r}.$ Vector $\vec{r}$ points from q towards the point where the electric field is calculated. For several charges the electric field will be equal to the vector sum of fields calculated for every charge.

For the continuous distribution of charge the electric field at point $\vec{x}$ is

${\displaystyle {\vec {E}}({\vec {x}})={1 \over 4\pi \varepsilon _{0}}\iiint \limits _{V}\,{\rho ({\vec {x}}')dV \over ({\vec {x}}'-{\vec {x}})^{2}}{\hat {\boldsymbol {r}}}'}$ , where $\rho(\vec{x'})$ is the charge density at point $\vec{x'},$ $\hat {\boldsymbol {r}}'$ is the unit vector pointing from $\vec{x}'$ to $\vec{x}$ . The integral is calculated over the whole volume V where the charge is distributed.