Question

Derive the formula for the intensity of electric field on the bisector (equatorial line) of an electric dipole.

Hint:

For an electric dipole, the electric field on the equatorial line is inversely proportional to the cube of the distance from the center of the dipole, and it is directed along the bisector.

Answer 1

Intensity of Electric Field on the Bisector of an Electric Dipole:
Consider an electric dipole consisting of two charges \( +q \) and \( -q \), separated by a distance \( 2a \). The dipole is placed along the \( x \)-axis, with the charges located at \( x = +a \) and \( x = -a \). The point where we need to calculate the electric field lies on the bisector (the equatorial line) of the dipole, which is at a distance \( r \) from the center of the dipole. The angle between the line joining the charges and the point on the bisector is \( 90^\circ \). To calculate the electric field at this point: 1. Electric field due to the positive charge \( +q \): The electric field due to a point charge is given by: \[ E = \frac{1}{4 \pi \varepsilon_0} \frac{q}{r^2}. \] For the positive charge at a distance \( r \) from the point, the electric field is directed radially away from the charge. Since the point lies on the bisector, the field due to the positive charge will have a component along the direction of the bisector. 2. Electric field due to the negative charge \( -q \): Similarly, the electric field due to the negative charge at the same distance \( r \) will also have a component along the bisector, but it will be directed towards the negative charge. 3. Net Electric Field on the Bisector: Since both electric fields are of equal magnitude but opposite directions, they add up along the bisector. The net electric field is: \[ E_{\text{net}} = \frac{1}{4 \pi \varepsilon_0} \frac{2p}{r^3}, \] where \( p = q \cdot 2a \) is the dipole moment of the system. Thus, the intensity of the electric field on the bisector (equatorial line) of an electric dipole is: \[ E_{\text{net}} = \frac{1}{4 \pi \varepsilon_0} \frac{2p}{r^3}. \]