A system of three charges are placed as shown in the figure: If $\mathrm{D} \gg \mathrm{d}$, the potential energy of the system is best given by:

(A) $\frac{1}{4 \pi \varepsilon_{0}}\left[-\frac{q^{2}}{d}-\frac{q Q d}{D^{2}}\right]$

(B) $\frac{1}{4 \pi \varepsilon_{o}}\left[-\frac{q^{2}}{d}-\frac{q Q d}{2 D^{2}}\right]$

(C) $\frac{1}{4 \pi \varepsilon_{o}}\left[-\frac{q^{2}}{d}+\frac{2 q Q d}{D^{2}}\right]$

(D) $\frac{1}{4 \pi \varepsilon_{o}}\left[+\frac{q^{2}}{d}-\frac{q Q d}{D^{2}}\right]$

JEE Main · 2019 · 09 Apr 2019 · Shift 1 · Electrostatics

A system of three charges are placed as shown in the figure:

If

D ≫ d

, the potential energy of the system is best given by:

(A) $\frac{1}{4 π ε _{0}} [- \frac{q ^{2}}{d} - \frac{q Q d}{D ^{2}}]$ ✓ Correct
(B) $\frac{1}{4 π ε _{o}} [- \frac{q ^{2}}{d} - \frac{q Q d}{2 D ^{2}}]$
(C) $\frac{1}{4 π ε _{o}} [- \frac{q ^{2}}{d} + \frac{2 q Q d}{D ^{2}}]$
(D) $\frac{1}{4 π ε _{o}} [+ \frac{q ^{2}}{d} - \frac{q Q d}{D ^{2}}]$

Answer: (A)

The energy has two parts. The close pair

+ q

and

- q

a distance

d

apart contributes the self-energy

\frac{1}{4 π ε _{0}} \frac{( - q ) ( q )}{d} = - \frac{1}{4 π ε _{0}} \frac{q ^{2}}{d}

. This pair behaves as a small dipole, and the distant charge

Q

D ≫ d

interacts with it through the dipole term, which falls off as

\frac{1}{D ^{2}}

and works out to

- \frac{1}{4 π ε _{0}} \frac{q Q d}{D ^{2}}

. Adding the two,

U = \frac{1}{4 π ε _{0}} [- \frac{q ^{2}}{d} - \frac{q Q d}{D ^{2}}]

JEE Main · 2019 · 09 Apr 2019 · Shift 1 · Single correct option