# Formula Continuous Charge Distribution in 2d Electric field Surface 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 origin to a location within the two-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 running from a point of the 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.

## Surface charge density

`$$ \sigma(\boldsymbol{r}) $$`

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

## Surface

`$$ A $$`Unit

`$$ \mathrm{m}^2 $$`

The surface of the considered two-dimensional charge distribution, which generates the electric field.

## Vacuum Permittivity

`$$ \varepsilon_0 $$`Unit

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

Vacuum permittivity is a physical constant and has the value \( \varepsilon_0 = 8.854 \cdot 10^{-12} \, \frac{\text{As}}{\text{Vm}} \).