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

Find the expression for the electric field intensity and the electric potential, due to a dipole at a point on the equatorial line. Would the electric field be necessarily zero at a point where the electric potential is zero? Give an example to illustrate your answer.

Accepted Answer

For the derivation of electric field intensity at the equatorial point of a dipole, Electric field at an equatorial point of a dipole. As shown in the figure, consider an electric dipole consisting of charges $- q$ and $+ q$ , separated by distance $2 a$ and placed in a vacuum. Let $P$ be a point on the equatorial line of the dipole at distance $r$ from it. i.e.,

Electric filed at point $P$ due to $+ q$ at point $P$ is

${\vec{E}}_{+ q} = \frac{1}{4 π ε_{0}} . \frac{q}{r^{2} - a^{2}}$ , directed along $B P$

Electric filed at point $P$ due to $- q$ charge is

${\vec{E}}_{- q} = \frac{1}{4 π ε_{0}} . \frac{q}{r^{2} + a^{2}}$ , directed along $P A$

Thus the magnitudes of ${\vec{E}}_{- q}$ and ${\vec{E}}_{+ q}$ are equal i.e.,

$E_{- q} = E_{+ q} \frac{1}{4 π ε_{0}} . \frac{q}{r^{2} + a^{2}}$

Clearly, the components of ${\vec{E}}_{- q}$ and ${\vec{E}}_{+ q}$ normal to the dipole axis will cancel out. The components parallel to the dipole axis add up. The total electric field ${\vec{E}}_{e q u a}$ is opposite to $\vec{P}$ .

$∴ {\vec{E}}_{e q u a} = - (E_{- q} \cos θ + E_{+ q} \cos θ) \hat{p}$

$= - 2 E_{- q} \cos θ \hat{p} [E_{- q} - E_{+ q}]$

$= - 2 . \frac{1}{4 π ε_{0}} . \frac{q}{r^{2} + a^{2}} . \frac{a}{\sqrt{r^{2} + a^{2}}} \hat{p}$

$[∵ \cos θ = \frac{a}{\sqrt{r^{2} + a^{2}}}]$

or ${\vec{E}}_{e q u a} = - \frac{1}{4 π ε_{0}} . \frac{p}{{(r^{2} + a^{2})}^{3 / 2}} \hat{p}$

where $p = 2 q a$ , is the electric dipole moment.

If the point $P$ is located far away from the dipole, $r ≫ a$ , then

${\vec{E}}_{e q u a} = - \frac{1}{4 π ε_{0}} . \frac{p}{r^{3}} \hat{p}$

Clearly, the direction of the electric field at any point on the equatorial line of the dipole will be antiparallel to the dipole moment $\vec{P}$

$E_{e q u a} = \frac{1}{4 π ε_{0}} \frac{p}{{(r^{2} + a^{2})}^{3 / 2}}$

Electric potential at an equatorial point a dipole.

As shown in the figure, consider an electric dipole consisting of charges $- q$ and $+ q$ and separated by distance $2 a$ . Let $P$ be a point on the perpendicular bisector of the dipole at distance $r$ from its centre $O$ .

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

$V = V_{1} + V_{2} = \frac{1}{4 π ε_{0}} . \frac{- q}{A P} + \frac{1}{4 π ε_{0}} . \frac{q}{B P}$

$= - \frac{1}{4 π ε_{0}} . \frac{q}{\sqrt{r^{2} + a^{2}}} + \frac{1}{4 π ε_{0}} . \frac{q}{\sqrt{r^{2} + a^{2}}} = 0$

$V_{e q u a} = 0$

No, the electric field may not be necessarily zero at a point where the electric potential is zero. For example, the electric potential at an equatorial point of a dipole is zero, while the electric field is not zero.

Find the expression for the electric field intensity and the electric potential, due to a dipole at a point on the equatorial line. Would the electric field be necessarily zero at a point where the electric potential is zero? Give an example to illustrate your answer.

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