Name: Equipotential Surfaces and Its Properties
Uploaded: 2019-07-19
Duration: 7 min 22 s
Description: The surface over which the electric potential is the same everywhere is called an equipotential surface. Let's more about it and its properties in this video.

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,

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

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).

Important Questions on Electric Potential

Learn and Prepare for any exam you want !

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).

Important Questions on Electric Potential

Learn and Prepare for any exam you want !

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).