Torque of a electric dipole in a uniform electric field derivation (12th physics notes Ch:-1)

 The torque of an electric dipole in a uniform electric field can be derived using the following steps:

1. Consider an electric dipole consisting of two charges +q and -q separated by a distance d, as shown below:

perl
              +q
               |
               |
          -------> E
               |
               |
              -q

2. The electric field E is assumed to be uniform and is applied perpendicular to the plane of the dipole.

3. The force acting on the positive charge +q is given by F = qE, and the force acting on the negative charge -q is given by F = -qE (since the electric field is applied in opposite directions for the two charges).

4. The net force on the dipole is the vector sum of the two forces:

r
          Fnet = F+ - F- = qE - (-qE) = 2qE

5. The torque τ of the dipole about its center is given by the vector product of the dipole moment p and the net force Fnet:

css
          τ = p x Fnet

6. The dipole moment p is defined as the product of the magnitude of either charge and the separation distance between them:

css
          p = qd

7. Substituting the expressions for p and Fnet into the equation for torque, we get:

scss
          τ = p x Fnet = (qd) x (2qE) = 2q^2dE sin θ

where θ is the angle between the dipole moment and the direction of the electric field.

8. Finally, we can express the torque in terms of the electric dipole moment p and the electric field strength E as:

c
          τ = p x E sin θ

This equation gives us the torque exerted on an electric dipole in a uniform electric field.