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

Derive the expression for the electric potential any point

P

, at distance

r

from the centre of an electric dipole, making angle

θ

, with its axis.

Accepted Answer

Consider an electric dipole consisting of two point charges $- q$ and $+ q$ and separated by distance $2 a$ , as shown in Fig. We need to determine the potential at a point $P$ at a distance $r$ from the centre $O$ , the direction $O P$ making an angle $θ$ with dipole moment $\vec{P}$ .

Lets assume $A P = r_{1}$ and $B P = r_{2}$

Net potential at point $P$ due to the dipole is

$V = V_{1} + V_{2}$

$V = \frac{1}{4 π ε_{0}} . \frac{- q}{r_{1}} + \frac{1}{4 π ε_{0}} . \frac{q}{r_{2}}$

$V = \frac{1}{4 π ε_{0}} [\frac{1}{r_{2}} - \frac{1}{r_{1}}]$

$V = \frac{1}{4 π ε_{0}} [\frac{r_{1} - r_{2}}{r_{1} r_{2}}]$

If the point $P$ lies far away from the dipole, then

$r_{1} - r_{2} ≃ A B \cos θ = 2 a \cos θ$ and $r_{1} r_{2} ≃ r^{2}$

$∴ V = \frac{1}{4 π ε_{0}} . \frac{2 a \cos θ}{r^{2}}$

or $V = \frac{1}{4 π ε_{0}} . \frac{p \cos θ}{r^{2}}$

$V = \frac{1}{4 π ε_{0}} \frac{\vec{p} . \vec{r}}{r^{3}} = \frac{1}{4 π ε_{0}} \frac{\vec{p} . \hat{r}}{r^{2}}$

Expression for the electric potential any point $P$ , at distance r from the centre of an electric dipole is $V = \frac{1}{4 π ε_{0}} \frac{\vec{p} . \hat{r}}{r^{2}}$

Derive the expression for the electric potential any point $P$ , at distance $r$ from the centre of an electric dipole, making angle $θ$ , with its axis.

Important Questions on Electrostatic Potential and Capacitance

Learn and Prepare for any exam you want !

Derive the expression for the electric potential any point P, at distance r from the centre of an electric dipole, making angle θ, with its axis.

Important Questions on Electrostatic Potential and Capacitance

Learn and Prepare for any exam you want !

Derive the expression for the electric potential any point $P$ , at distance $r$ from the centre of an electric dipole, making angle $θ$ , with its axis.