Question

derive time period formula of a charge in shm between two idnetical charges both up and along motion

Answer 1

T = 2\pi\sqrt{\frac{I}{qEl}}

Answer 2

Solution Explanation For a dipole consisting of two identical charges (each of charge $q$) separated by a distance $l$, the net dipole moment is $p = q\,l$. When placed in a uniform electric field $E$, for a small angular displacement $\theta$ the restoring torque is

$$\tau = -pE\theta = -qEl\,\theta.$$

Using the rotational equation of motion

$$I\frac{d^2\theta}{dt^2} = -qEl\,\theta,$$

we identify the angular frequency for SHM as

$$\omega = \sqrt{\frac{qEl}{I}}.$$

Thus, the time period $T$ is

$$T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{I}{qEl}}.$$

Answer

$$T = 2\pi\sqrt{\frac{I}{qEl}}$$

(derived for both 'up' (oscillations about the center) and 'along' (oscillations about one end) motions)