Name: Electric Energy Density
Uploaded: 2019-07-19
Duration: 4 min 11 s
Description: There is an energy associated with the electrostatic field. The expression of this electric field energy density is derived using the energy stored in a para...

Question

Obtain the expression for the potential due to a point charge.

Accepted Answer

Electric potential is defined as the amount of work done in bringing a unit positive charge from a reference point to the specific point against the electric field.

Consider a positive point charge $q$ placed at the origin $O$ . We need to calculate electric potential at a point $P$ at distance $r$ from charge $q$ , as shown in. By definition, the electric potential at point $P$ will be equal to the amount of work done in bringing a unit positive charge from infinity to the point $P$ .

Suppose a test charge $q_{0}$ is placed at point $A$ at distance $x$ from $O$ . By Coulomb's law, the electrostatic force acting on the charge $q_{0}$ is

$F = \frac{1}{4 π ε_{0}} . \frac{q q_{0}}{x^{2}}$

The direction of the force $\vec{F}$ will act away from the charge $q$ . The small work done in moving the test charge $q_{0}$ from $A$ to $B$ through small displacement $\vec{d x}$ against the electrostatic force is

$\vec{d W} = \vec{F} . \vec{d x}$

$d W = F d x \cos 180^{o}$ , Where angle between the direction of force and displacement is $180 °$

$d W = - F \times d x$

The total work done in moving the charge $q_{0}$ from infinity to the point $P$ will be

$W = \int d W = - \int_{\infty}^{r} F d x = - \int_{\infty}^{r} \frac{1}{4 π ε_{0}} . \frac{q q_{0}}{x^{2}} d x$

$W = - \frac{q q_{0}}{4 π ε_{0}} \int_{\infty}^{r} x^{- 2} d x = - \frac{q q_{0}}{4 π ε_{0}} {[- \frac{1}{x}]}_{\infty}^{r}$

$W = \frac{q q_{0}}{4 π ε_{0}} [\frac{1}{r} - \frac{1}{\infty}] = \frac{1}{4 π ε_{0}} . \frac{q q_{0}}{r}$

Hence the work done in moving a unit test charge from infinity to the $P$ is

$V = \frac{W}{q_{0}}$ or

$V = \frac{1}{4 π ε_{0}} . \frac{q}{r}$

Obtain the expression for the potential due to a point charge.

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