Name: Energy Density in a Capacitor
Uploaded: 2019-07-19
Duration: 3 min 1 s
Description: The energy in the capacitor is stored in the form of the electric field. Let us derive a mathematical formula to calculate this energy in terms of the electr...

Question

Derive an expression for the electric potential at any point at a distance $r$ from the centre of an electric dipole. Hence find the potential if the point lies on

(i) axial line and (ii) equatorial line

Accepted Answer

An electric dipole is defined as a couple of opposite charge $q$ and $- q$ separated by distance.

Let an electric field consist of two equal and opposite point charges $- q$ at $A$ and $+ q$ at $B$ separated by a mall distance $A B = 2 a$ , with centre at $O$ .

The dipole moment, $p = q \times 2 a$

We will calculate potential at any point $P$ , where

$O P = r$ and $∠ B O P = θ$

Let $B P = r_{1}$ and $A P = r_{2}$

Draw $A C$ perpendicular $P Q$ and $B D$ perpendicular $P O$

In $△ A O C$

$\cos (θ) = \frac{O C}{a}$

$O C = a \cos (θ)$

Similarly, $O D = a \cos (θ)$

Potential at $P$ due to $+ q = \frac{1}{4 {πε}_{0}} \frac{q}{r_{2}}$

And potential at $P$ due to $- q = - \frac{1}{4 {πε}_{0}} \frac{q}{r_{1}}$

Net potential at $P$ due to the dipole

$V = \frac{1}{4 {πε}_{0}} (\frac{q}{r_{2}} - \frac{q}{r_{1}})$

$V = \frac{q}{4 {πε}_{0}} (\frac{1}{r_{2}} - \frac{1}{r_{1}})$

Now, $r_{1} = A P = C P$

$O P + O C = r + a \cos (θ)$

$r_{2} = B P = D P$

$O P - O D = r - a \cos (θ)$

$V = \frac{q}{4 {πε}_{0}} (\frac{1}{r - a \cos (θ)} - \frac{1}{r + a \cos (θ)})$

$V = \frac{q}{4 {πε}_{0}} (\frac{2 a \cos (θ)}{r^{2} - a^{2} \cos^{2} (θ)})$

$∵$ Dipole moment $P = q (2 a)$

$V = \frac{K P \cos (θ)}{r^{2} - a^{2} \cos^{2} (θ)}$

(i) When the point $P$ lies on the axial line of the dipole, $θ = 0^{°}$

$\cos (θ) = 1$

$V = \frac{K P}{r^{2} - a^{2}}$

If $a < < r, V = \frac{K P}{r^{2}}$

Thus, due to an electric dipole moment, potential $V \propto \frac{1}{r^{2}}$

(ii) when the point lies on the equatorial line of the dipole $θ = 90^{°}$ , $\cos (θ) = 0$

Therefore, electrical potential due to an electric dipole is zero at every point on the equatorial line of the dipole.

Derive an expression for the electric potential at any point at a distance $r$ from the centre of an electric dipole. Hence find the potential if the point lies on

(i) axial line and (ii) equatorial line

Important Questions on Electrostatic Potential and Capacitance

Learn and Prepare for any exam you want !

Derive an expression for the electric potential at any point at a distance r from the centre of an electric dipole. Hence find the potential if the point lies on (i) axial line and (ii) equatorial line

Important Questions on Electrostatic Potential and Capacitance

Learn and Prepare for any exam you want !

Derive an expression for the electric potential at any point at a distance $r$ from the centre of an electric dipole. Hence find the potential if the point lies on

(i) axial line and (ii) equatorial line