Question

(a) Depict the equipotential surfaces for a system of two identical positive point charges placed a distance ‘d’ apart. 

(b) Deduce the expression for the potential energy of a system of two point charges q₁ and q₂ brought from infinity to the points vector r₁ and vector r₂ respectively in the presence of external electric field vector E.

Answer 1

(a) Equipotential surfaces due to two identical charges is shown in fig.

(b) Potential energy of a system of two charges in an external electric field. Suppose q₁  and q₂  are two charges brought from infinity at locations vector r₁ and vector r₂ respectively in an external electric field. 

Let V(vector r₁) and V(vector r₂) be the potentials at positions vector r₁ and vector r₂ due to external electric field vector E. In this case work is done in bringing charges q₁  and q₂  against their own electric fields and external electric fields. 

Work done in bringing charge q₁  from ∝ at location vector r₁ is W₁ = q₁V (vector r₁)

Work done in bringing q₂  against the electric field at location vector r₂  is W₂ = q₂V(vector r₂)

Work done on q₂  against the electric field due to q₁  is

∴ Potential energy of system = Work done in assembling the configuration