Question

derive general formula for time period of a charge with shm in a dipole, for both positive negative charge and up and along motion

Answer 1

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

Answer 2

Solution Explanation We start by recalling that for an electric dipole (of moment   $p = q\,l$ ) in a uniform electric field $E$, the torque experienced is   $\tau = -p E \sin\theta$. For small angular displacements (i.e. when $\theta$ is small) we use the approximation   $\sin\theta\approx\theta$ so that the equation becomes   $\tau \approx -p E \theta$.

The rotational equation of motion is given by   $I\,\dfrac{d^2\theta}{dt^2} = -p E\,\theta$, which is the standard form for simple harmonic motion with angular frequency   $\omega=\sqrt{\dfrac{pE}{I}}$.

Thus the general time period of oscillation is

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

If the dipole consists of two charges (each of mass $m$) separated by a distance $l$, the moment of inertia $I$ depends on the axis of rotation. Two common cases are:

1. Oscillations about the center (symmetric “up‐down” motion):  For two point masses (each of mass $m$) located at a distance $l/2$ from the center,   $I = m\left(\frac{l}{2}\right)^2 + m\left(\frac{l}{2}\right)^2 = \frac{ml^2}{2}.$  Thus the time period is   $T = 2\pi \sqrt{\frac{ml^2/2}{qEl}} = 2\pi\sqrt{\frac{ml}{2qE}}.$

2. Oscillations about one end (“pivoted” or “along” motion):  If one charge is fixed and the other (of mass $m$) oscillates about that pivot, then   $I = m\,l^2.$  Thus the time period is   $T = 2\pi\sqrt{\frac{ml^2}{qEl}} = 2\pi\sqrt{\frac{ml}{qE}}.$

Notice that the sign of the charge (whether positive or negative) does not affect the period, since only the magnitude $q$ appears in the formulas.

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Final Answer The general formula for the time period of small oscillations of a dipole in a uniform electric field is

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

where • For oscillation about its center (with two equal masses $m$):   $T = 2\pi\sqrt{\frac{ml}{2qE}},$ • For oscillation about one end (one mass $m$ pivoted):   $T = 2\pi\sqrt{\frac{ml}{qE}}.$

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Additional Information

• Subject: Physics
• Chapter: Electric Charges and Fields (NCERT Class 12) and Rotational Motion
• Topic: Electric Dipoles in Uniform Fields, Simple Harmonic Motion (SHM)
• Difficulty Level: Medium
• Question Type: descriptive

This derivation is suitable for a 12th-grade student and uses basic approximations and rotational dynamics to arrive at the result.