Formula: Charge in Radial E-field Energy Electric charge Distance
$$W_{\text{pot}} ~=~ \frac{ \class{red}{Q} }{4\pi \, \varepsilon_0} \, \frac{ \class{red}{q} }{r}$$
$$W_{\text{pot}} ~=~ \frac{ \class{red}{Q} }{4\pi \, \varepsilon_0} \, \frac{ \class{red}{q} }{r}$$
$$\class{red}{q} ~=~ \frac{4\pi \, \varepsilon_0 \, W_{\text{pot}} \, r}{\class{red}{Q}}$$
$$\class{red}{Q} ~=~ \frac{4\pi \, \varepsilon_0 \, W_{\text{pot}} \, r}{\class{red}{q}}$$
$$r ~=~ \frac{\class{red}{Q} \, \class{red}{q}}{4\pi \, \varepsilon_0 \, W_{\text{pot}}}$$
Potential energy
$$ W_{\text{pot}} $$ Unit $$ \mathrm{J} = \mathrm{Nm} = \frac{ \mathrm{kg} \, \mathrm{m^2} }{ \mathrm{s}^2 } $$
Potential energy of a charged particle with charge \(\class{red}{q}\) in an external electric field. The charge \(\class{red}{q}\) is at a distance \(r\) from the source charge \(\class{red}{Q}\) which generates the electric field.
Electric charge
$$ q $$ Unit $$ \mathrm{C} = \mathrm{As} $$
Electric charge of the particle for which the potential energy is calculated. To bring a positive charge from infinity to the point \(P\) to the positive charge \(\class{red}{Q}\), energy must be invested.
Source charge
$$ \class{red}{Q} $$ Unit $$ \mathrm{C} = \mathrm{As} $$
Electric charge that generates the electric field.
Distance
$$ r $$ Unit $$ \mathrm{m} $$
Distance of the small charge \(\class{red}{q}\) from the source charge \(\class{red}{Q}\).
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:
$$ \varepsilon_0 ~\approx~ 8.854 \, 187 \, 8128 ~\cdot~ 10^{-12} \, \frac{\mathrm{As}}{\mathrm{Vm}} $$