A dipole comprises of two charged particles of identical magnitude q and opposite in nature. The mass ‘m’ of the positive charged particle is half of the mass of the negative charged particle. The two charges are separated by a distance ‘l’. If the dipole is placed in a uniform electric field ‘$\overrightarrow{E}$; such a way that dipole axis makes a very small angle with the electric field, ‘$\overrightarrow{E}$'. The angular frequency of the oscillations of the dipole when released is given by.

Question

A thin half ring of radius $35 \text{ cm}$ is uniformly charged with a total charge of $Q$ coulomb. If the magnitude of the electric field at centre of the half ring is $100 \text{ V/m}$, then the value of $Q$ is ______ $\text{nC}$.
($\epsilon_o = 8.85 \times 10^{-12} \text{ C}^2/\text{Nm}^2$ and $\pi = 3.14$)

Answer 1

To solve this problem, we need to find the angular frequency of oscillation for the dipole in a uniform electric field. Let's go through the solution step-by-step:

Consider a dipole made of two charges $ q $ and $-q$, separated by a distance $ l $. The masses of the charges are $ m $ and $ 2m $ respectively. The dipole is placed in a uniform electric field $\overrightarrow{E}$.
The torque ($\tau$) acting on the dipole in the electric field is given by the formula: $\tau = pE \sin \theta$ , where $ p = ql $ is the dipole moment and $\theta$ is the angle between the dipole axis and the electric field.
For small oscillations, $\sin \theta \approx \theta$, so the torque becomes: $\tau = pE \theta$ .
Using the relation $\tau = I \alpha$, where $ I $ is the moment of inertia of the dipole and $\alpha$ is the angular acceleration, we can write: $ pE \theta = I \frac{d^2\theta}{dt^2} $ .
The moment of inertia $ I $ for point masses at a distance can be calculated as: \[ I = m \left(\frac{l}{2}\right)^2 + (2m) \left(\frac{l}{2}\right)^2 = \frac{ml^2}{4} + \frac{2ml^2}{4} = \frac{3ml^2}{4} \]
Substitute the value of $ I $ into the torque equation: \[ pE \theta = \frac{3ml^2}{4} \frac{d^2 \theta}{dt^2} \]
Rearranging gives: \[ \frac{d^2 \theta}{dt^2} + \frac{4qE}{3ml} \theta = 0 \]
This equation is of the form of simple harmonic motion (SHM), with angular frequency $\omega$ given by: \[ \omega = \sqrt{\frac{4qE}{3ml}} \]
Therefore, the angular frequency of oscillations of the dipole is: \[ \boxed{\sqrt{\frac{4qE}{3ml}}} \]

The correct option is: $\sqrt{\frac{4qE}{3ml}}$

All other options are incorrect as they do not match with the obtained formula for the angular frequency of the oscillating dipole in the field given the specified conditions and parameters.

Answer 2

To solve this problem, we need to find the angular frequency of oscillation for the dipole in a uniform electric field. Let's go through the solution step-by-step:

Consider a dipole made of two charges $ q $ and $-q$, separated by a distance $ l $. The masses of the charges are $ m $ and $ 2m $ respectively. The dipole is placed in a uniform electric field $\overrightarrow{E}$.
The torque ($\tau$) acting on the dipole in the electric field is given by the formula: $\tau = pE \sin \theta$ , where $ p = ql $ is the dipole moment and $\theta$ is the angle between the dipole axis and the electric field.
For small oscillations, $\sin \theta \approx \theta$, so the torque becomes: $\tau = pE \theta$ .
Using the relation $\tau = I \alpha$, where $ I $ is the moment of inertia of the dipole and $\alpha$ is the angular acceleration, we can write: $ pE \theta = I \frac{d^2\theta}{dt^2} $ .
The moment of inertia $ I $ for point masses at a distance can be calculated as: \[ I = m \left(\frac{l}{2}\right)^2 + (2m) \left(\frac{l}{2}\right)^2 = \frac{ml^2}{4} + \frac{2ml^2}{4} = \frac{3ml^2}{4} \]
Substitute the value of $ I $ into the torque equation: \[ pE \theta = \frac{3ml^2}{4} \frac{d^2 \theta}{dt^2} \]
Rearranging gives: \[ \frac{d^2 \theta}{dt^2} + \frac{4qE}{3ml} \theta = 0 \]
This equation is of the form of simple harmonic motion (SHM), with angular frequency $\omega$ given by: \[ \omega = \sqrt{\frac{4qE}{3ml}} \]
Therefore, the angular frequency of oscillations of the dipole is: \[ \boxed{\sqrt{\frac{4qE}{3ml}}} \]

The correct option is: $\sqrt{\frac{4qE}{3ml}}$

All other options are incorrect as they do not match with the obtained formula for the angular frequency of the oscillating dipole in the field given the specified conditions and parameters.

The Correct Option is A

Solution and Explanation

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