Question

How does the potential energy of

1. Mutual interaction
2. TwoNet electrostatic charges

​change when they are placed in an external electric field.

Answer 1

Consider two charges q₁ and q₂ located at distance r₁ and r₂ respectively in an external electric field . In this situation first we calculate the work done in bringing the charge q₁ from infinity to r₁ against the external field i.e W₁ =q₁ V (r₁)
Similarly for charge q₂ , W₂ = q₂ V(r₂)
In this case work is done not only against the external field E but also against the field due to q_{1
${\mathit{W}}_{\mathbf{12}}\mathbf{}\mathbf{=}\mathbf{}\frac{{\mathbf{q}}_{\mathbf{1}}{\mathbf{q}}_{\mathbf{2}}}{\mathbf{4}\mathbf{\pi }{\mathbf{\epsilon }}_{\mathbf{o}}\mathbf{}{\mathbf{r}}_{\mathbf{12}}}$
Hence Net potential energy of the system = the total work done in assembling the configuration
= ${\mathit{W}}_{\mathbf{1}\mathbf{}}\mathbf{+}{\mathit{W}}_{\mathbf{2}\mathbf{}}\mathbf{+}\mathbf{}{\mathit{N}}_{\mathbf{12}}$}