Question

Consider the configuration of a system of four charges each of value $+ q$ . Find the work done by external agent in changing the configuration of the system from figure (a) to figure (b).

Accepted Answer

In the first figure, we have a rectangle configuration.

Length of square side $= a$

In the second figure, we have a circle configuration.

Radius of circle $= a$

When we place charges in square configuration, the Potential energy of the system will be

$U_{i} = U_{12} + U_{23} + U_{34} + U_{41} + U_{13} + U_{24}$

We can see from the figure above that

$U_{12} = U_{23} = U_{34} = U_{41}$ and $U_{13} = U_{24}$

Potential energy, $U_{i} = 4 U_{12} + 2 U_{13}$

$U_{12} = \frac{k q^{2}}{a} U_{13} = \frac{k q^{2}}{(\sqrt{2} a)}$

Substituting these values in above expression, we get,

$\Rightarrow U_{i} = 4 (\frac{k q^{2}}{a}) + 2 (\frac{k q^{2}}{\sqrt{2} a}) = \frac{4 k q^{2}}{a} + \frac{\sqrt{2} k q^{2}}{a} = \frac{k q^{2}}{a} (4 + \sqrt{2})$

When we place charges in circle configuration, the potential energy of the system,

$U_{f} = U_{a b} + U_{b c} + U_{c d} + U_{d a} + U_{a c} + U_{b d}$

We can see from the figure above,

$U_{a b} = U_{b c} = U_{c d} = U_{d a}$ and $U_{a c} = U_{b d}$

Potential energy, $U_{f} = 4 U_{a b} + 2 U_{a c}$

$U_{a b} = \frac{k q^{2}}{\sqrt{2} a} U_{a c} = \frac{k q^{2}}{2 a}$

Substituting the values above, to get potential energy,

$U_{f} = 4 (\frac{k q^{2}}{\sqrt{2} a}) + 2 (\frac{k q^{2}}{2 a}) = \frac{4 k q^{2}}{\sqrt{2} a} + \frac{k q^{2}}{a} = \frac{k q^{2}}{a} (2 \sqrt{2} + 1)$

Work done in changing the configuration $=$ Final Potential energy $-$ Initial Potential energy

$W = U_{f} - U_{i}$

Substituting the values from above, we get,

$W = \frac{k q^{2}}{a} (2 \sqrt{2} + 1) - \frac{k q^{2}}{a} (4 + \sqrt{2}) = \frac{k q^{2}}{a} (2 \sqrt{2} + 1 - 4 - \sqrt{2}) = \frac{k q^{2}}{a} (\sqrt{2} - 3) W = \frac{q^{2}}{4 π ε_{0} a} (\sqrt{2} - 3)$

B M Sharma Solutions for Chapter: Electric Potential, Exercise 3: Concept Application Exercise

B M Sharma Physics Solutions for Exercise - B M Sharma Solutions for Chapter: Electric Potential, Exercise 3: Concept Application Exercise

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