Name: Equipotential Surfaces and Its Properties
Uploaded: 2019-07-19
Duration: 7 min 22 s
Description: The surface over which the electric potential is the same everywhere is called an equipotential surface. Let's more about it and its properties in this video.

Question

Define electric potential. Derive an expression for electric potential due to an electric dipole and hence electric potential at a point on the equatorial line of electric dipole.

Accepted Answer

Electric potential is the amount of work needed to move a unit charge from a reference point to a specific point against an electric field.

Let an electric dipole consist of two equal and opposite point charges $- q$ at $A$ and $+ q$ at $B$ , separated by a small 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 $△ A O C, \cos θ = \frac{O C}{O A} = \frac{O C}{a}$

$O C = a \cos θ$

Similarly, $O D = a \cos θ$

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

Potential at P due to $- q = \frac{1}{4 π ε_{0}} \frac{q}{r_{1}}$

Net potential at P due to the dipole $V = \frac{1}{4 π ε_{0}} (\frac{q}{r_{2}} - \frac{q}{r_{1}}) = \frac{q}{4 π ε_{0}} (\frac{1}{r_{2}} - \frac{1}{r_{1}})$

$r_{1} = A P = C P = O P + O C = r + a \cos θ$

$r_{2} = B P = D P = O P - O D = r - a \cos θ$

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

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

$V = \frac{p}{4 π ε_{0}} (\frac{\cos θ}{r^{2} - a^{2} \cos^{2} θ})$ $[∵ p = q \times 2 a]$

When the point $P$ lies on the equatorial line of the dipole, $θ = 90 °$

$V = \frac{p}{4 π ε_{0}} (\frac{0}{r^{2} - a^{2} \times 0}) = 0$

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

Define electric potential. Derive an expression for electric potential due to an electric dipole and hence electric potential at a point on the equatorial line of electric dipole.

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