Question

Potential energy of a system comprising of point charges is U1. When a charge q is added in the system without disturbing other charges, the potential energy becomes U2. The potential of the point where the charge q is placed in the system is U2-U

Answer 1

Given :

Initial potential energy of the system = U₁

Final potential energy of the system = U₂

To Find :

The potential of the point where the charge 1 is placed

Solution :

• Let us assume initially the charges q₁ and q₂ are in the system and the charge q is brought into the system from infinity

• The initial potential energy of the system is

                  $U_{1} =\frac{1}{4\pi \epsilon } \frac{q_{1}q_{2}}{r_{1} }$

• The potential energy of the sysytem when charge q is introduced

                 $U_{2} =\frac{1}{4\pi \epsilon } \frac{q_{1}q_{2}}{r_{1} }+ \frac{1}{4\pi \epsilon } \frac{q_{1}q}{r_{2} }+ \frac{1}{4\pi \epsilon } \frac{qq_{2}}{r_{3} }$

                 $U_{2} =U_{1} +q( \frac{1}{4\pi \epsilon } \frac{q_{2}}{r_{1} }+ \frac{1}{4\pi \epsilon } \frac{q_{1}}{r_{1} })$

                 $U_{2} =U_{1} +qV$

                 $V=\frac{U_{2} -U_{1} }{q}$

The potential energy of the system is  (U₂-U₁)/q .