Single Choice

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 (i) to figure (ii).

A$$k \dfrac{q^2}{a} (3 + \sqrt{2})$$
B$$-k \dfrac{q^2}{a} (3 - \sqrt{2})$$
Correct Answer
C$$k \dfrac{q^2}{a} (2\sqrt{2} + 1)$$
D$$k \dfrac{q^2}{a} (3 - \sqrt{2})$$

Solution

The work done by the external agent will be the difference of the potential energy.

P.E of the Ist configuration,
$$U_{1} = U_{12} + U_{23} +U_{13} + U_{14} + U_{24} + U_{34}$$

or, $$U_{1} = kq^2 \left \{ \dfrac{1}{a} + \dfrac{1}{a} + \dfrac{1}{\sqrt{2} a} + \dfrac{1}{a} + \dfrac{1}{\sqrt{2} a} + \dfrac{1}{a} \right \}$$

or, $$U_{1} = kq^2 \left \{ \dfrac{4}{a} + \dfrac{\sqrt{2}}{a} \right \}$$

or, $$U_{1} = k \dfrac{q^2}{a} \{4 + \sqrt{2} \}$$

P.E of the second configuration,
$$U_{2} = U_{12} + U_{23} + U_{13} + U_{14} + U_{24} + U_{34}$$

or, $$U_{2} = kq^2 \{ \dfrac{1}{\sqrt{2} a} + \dfrac{1}{\sqrt{2} a} + \dfrac{1}{2a} + \dfrac{1}{\sqrt{2} a} + \dfrac{1}{2a} + \dfrac{1}{\sqrt{2} a} \}$$

or, $$U_{2} = k \dfrac{q^2}{a} \{2 \sqrt{2} + 1\}$$

Work done by the external agent,
$$W_{\text{ext}} = U_{2} - U_{1} = k \dfrac{q^2}{a} \{2 \sqrt{2} + 1 - 4 - \sqrt{2} \}$$

or, $$W_{\text{ext}} = k \dfrac{q^2}{a} \{ \sqrt{2} - 3\}$$

or, $$W_{\text{ext}} = -k \dfrac{q^2}{a} \{3 - \sqrt{2} \}$$

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