# Formula Continuous Charge Distribution in 3d Electric field Charge density

## Electric field

`$$ \boldsymbol{E}(\boldsymbol{R}) $$`Unit

`$$ \frac{\mathrm V}{\mathrm m} $$`

The E-field indicates how large and in which direction the electric force at the location \(\boldsymbol{R}\) would act on a test charge if it were placed at the location under consideration.

## Position vector

`$$ \boldsymbol{r} $$`Unit

`$$ \mathrm{m} $$`

It goes from the coordinate origin to a location within the three-dimensional charge distribution.

## Field vector

`$$ \boldsymbol{R} $$`Unit

`$$ \mathrm{m} $$`

It goes from the origin to the location (field point) where the electric field is to be calculated.

The connecting vector \(\boldsymbol{R} - \boldsymbol{r}\) is the vector going from a point of charge distribution \(\boldsymbol{r}\) to the considered field point \(\boldsymbol{R}\). Here \(\frac{\boldsymbol{R} ~-~ \boldsymbol{r}}{|\boldsymbol{R} ~-~ \boldsymbol{r}|}\) is the unit vector of the connecting vector.

## Volume

`$$ V $$`Unit

`$$ \mathrm{m}^3 $$`

The volume of the considered three-dimensional charge distribution, which generates the electric field.

## Charge density

`$$ \rho(\boldsymbol{r}) $$`Unit

`$$ \frac{\mathrm{C}}{\mathrm{m}^3} $$`

Charge per volume at location \(\boldsymbol{r}\) within the considered three-dimensional charge distribution.

## Vacuum Permittivity

`$$ \varepsilon_0 $$`Unit

`$$ \frac{\mathrm{As}}{\mathrm{Vm}} $$`

The vacuum permittivity is a physical constant that appears in equations involving electromagnetic fields. It has the following experimentally determined value:

`$$ \varepsilon_0 ~\approx~ 8.854 \, 187 \, 8128 ~\cdot~ 10^{-12} \, \frac{\mathrm{As}}{\mathrm{Vm}} $$`