Two equal point charges are fixed at x = –a and x = +a on the axis. Another point charge Q is placed at the origin. The change in the electrical potential energy of system, when it is displaced by a small distance x along the X-axis is approximately proportional to

A
$x$
B
$x^{2}$
C
$x^{3}$
D
$\frac{1}{x}$

Solution:

$\begin{matrix} U_{1} = \frac{1}{4 π \in_{0}} [\frac{q^{2}}{2 a} + \frac{q Q}{a} + \frac{q Q}{a}] \\ U_{2} = \frac{1}{4 π \in_{0}} [\frac{q^{2}}{2 a} + \frac{q Q}{a - x} + \frac{q Q}{a + x}] \\ Δ U = U_{2} - U_{1} \\ = \frac{Q q}{4 π ε_{0}} [\frac{1}{a - x} + \frac{1}{a + x} - \frac{2}{a}] \\ = \frac{Q q}{4 π ε_{0}} [\frac{2 x^{2}}{a^{3}}] \end{matrix}$

Two equal point charges are fixed at x = –a and x = +a on the axis. Another point charge Q is placed at the origin. The change in the electrical potential energy of system, when it is displaced by a small distance x along the X-axis is approximately proportional to

Solution:

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