Question

Three identical charges each of $2 \,\mu C$ are placed at the vertices of a triangle $A B C$ as shown in the figure. If $A B+A C=12\, cm$ and $A B \cdot A C=32\, cm ^{2}$, the potential energy of the charge at $A$ is

• A. 1.53 J
• B. 5.31 J
• C. 3.15 J
• D. 1.35 J

Answer 1

Answer: (D)

Given each charge, $q=2 \,\mu C$
or $q=2 \times 10^{-6} C$
$A B+A C=12 \,cm \dots$(i)
$A B \cdot A C=32\, cm ^{2}\dots$ (ii)
From equation (i), $A B=12-A C$
Put the value of $A B$ in equation (ii)
$(12-A C) A C=32$ or $ A C^{2}-12 A C+32=0$
On solving, we get $A C=8$ or $4$
$\therefore $ If $AC = 8\,cm, AB = 4\,cm$ or
$AB = 0.04\,m, AC = 0.08\,m$
The $P . E$ of the charge at $A$ is
$U=\frac{1}{4 \pi \varepsilon_{0}}\left[\frac{q^{2}}{A B}+\frac{q^{2}}{A C}\right]$
$=9 \times 10^{9}\left[\frac{4 \times 10^{-12}}{0.04}+\frac{4 \times 10^{-12}}{0.08}\right]=1.35 \,J$