Name: Electric and Gravitational Forces
Uploaded: 2019-07-19
Duration: 3 min 11 s
Description: Do you know the difference and similarities between the gravitational force and electrostatic force? Watch this video to explore it. Don't miss it.

Question

Define the term 'electric dipole moment'. Is it a scalar or vector? Deduce an expression for the electric field at a point on the equatorial plane of an electric dipole of length

2 a

.

Accepted Answer

Electric dipole moment is a vector quantity. Its magnitude is the product of either charge $q$ and the separation between them and its direction is from negative charge to positive charge.

$\vec{p} = q \times 2 \vec{a}$

Here $2 \vec{a}$ is the vector length from negative charge to positive charge.

Its SI unit is coulomb metre ( $C m$ ).

Consider a point $P$ on the equatorial line of an electric dipole as shown in the figure.

The electric field at $P$ due to $+ q$ charge ${\vec{E}}_{+ q} = \frac{1}{4 π ε_{0}} \frac{q}{r^{2} + a^{2}}$ , along $B P$

The electric field at $P$ due to $- q$ charge ${\vec{E}}_{- q} = \frac{1}{4 π ε_{0}} \frac{q}{r^{2} + a^{2}}$ , along $P A$

As the magnitudes of ${\vec{E}}_{+ q}$ and ${\vec{E}}_{- q}$ are equal, hence, components of ${\vec{E}}_{+ q}$ and ${\vec{E}}_{- q}$ perpendicular to the axis of the dipole cancel each other. Resultant ${\vec{E}}_{+ q}$ and ${\vec{E}}_{- q}$ is the sum of components of ${\vec{E}}_{+ q}$ and ${\vec{E}}_{- q}$ parallel to the axis of the dipole.

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

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

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

$= - \frac{1}{4 π ε_{0}} \frac{p}{{(r^{2} + a^{2})}^{3 / 2}} \hat{p}$

Here $p = q \times 2 a$ is the dipole moment of the dipole.

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

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

From the above equation, we can conclude that the direction of the electric field on the equatorial line is opposite to the direction of the dipole moment.

Define the term 'electric dipole moment'. Is it a scalar or vector? Deduce an expression for the electric field at a point on the equatorial plane of an electric dipole of length $2 a$ .

Important Questions on Electric Charges and Field

Learn and Prepare for any exam you want !

Define the term 'electric dipole moment'. Is it a scalar or vector? Deduce an expression for the electric field at a point on the equatorial plane of an electric dipole of length 2a.

Important Questions on Electric Charges and Field

Learn and Prepare for any exam you want !

Define the term 'electric dipole moment'. Is it a scalar or vector? Deduce an expression for the electric field at a point on the equatorial plane of an electric dipole of length $2 a$ .