Question

Obtain the formula for the electric potential on the axial line of an electric dipole.

Hint:

The electric potential on the axial line of a dipole is inversely proportional to the square of the distance from the dipole.

Answer 1

Step 1: Electric potential of a dipole.
The electric potential at a point on the axial line of an electric dipole is given by: \[ V = \frac{1}{4 \pi \epsilon_0} \frac{2p \cos \theta}{r^2} \] where: - \( V \) is the electric potential, - \( p \) is the dipole moment (\( p = q \times 2a \), where \( q \) is the charge and \( 2a \) is the separation between the charges), - \( r \) is the distance from the center of the dipole to the point where the potential is being calculated, - \( \theta \) is the angle between the dipole axis and the position vector of the point, - \( \epsilon_0 \) is the permittivity of free space.
Step 2: Electric potential along the axial line.
For points along the axial line, \( \theta = 0 \), so \( \cos 0 = 1 \), and the electric potential simplifies to: \[ V = \frac{1}{4 \pi \epsilon_0} \frac{2p}{r^2} \]
Step 3: Conclusion.
The electric potential at a point on the axial line of an electric dipole is: \[ V = \frac{1}{4 \pi \epsilon_0} \frac{2p}{r^2} \]