Problem 8 - A Complete Resource Book in Physics for JEE Main

Question

A simple pendulum of length \(l\) has a bob of mass \(m\), with a charge \(q\) on it. A vertical sheet of charge, with surface charge density \(\sigma\) passes through the point of suspension. At equilibrium, the string makes an angle \(\theta\) with the vertical, then (A) \(\tan \theta=\frac{\sigma q}{2 \varepsilon_{0} m g}\) (B) \(\tan \theta=\frac{\sigma q}{\varepsilon_{0} m g}\) (C) \(\cot \theta=\frac{\sigma q}{2 \varepsilon_{0} m g}\) (D) \(\cot \theta=\frac{\sigma q}{\varepsilon_{0} m g}\)

Step-by-Step Solution

Answer

The correct equation is \( \tan \theta = \frac{\sigma q}{2 \varepsilon_{0} m g} \), so option (A) is the correct answer.

Step 1: Identify the Forces

There are three sources of forces applied on the bob: the electrostatic force \(F_q\) which is horizontal, the weight \(W = m \cdot g\) which is vertical, and the tension \(T\) in the string. The angle theta is formed by the vertical axis and the string. In equilibrium, as per Newton's second law of motion, these forces equate to zero, so \(T - W \cdot \cos \theta - F_q \cdot \sin \theta = 0\) and \(W \cdot \sin \theta - F_q \cdot \cos \theta = 0\).

Step 2: Information about the electrostatic force

The electrostatic force \(F_q\) is due to the vertical sheet of charge and is given by \(F_q = \frac{\sigma \cdot q}{2 \cdot \varepsilon_0}\), where \(\sigma\) is the charge density, \(q\) is the charge on the bob, and \(\varepsilon_0\) is the permittivity of free space.

Step 3: Equating the Forces for Equilibrium

From 'Step 1', the component of \(W\) along the string counteracts the component of \(F_q\) perpendicular to the string. Similarly, the component of \(W\) perpendicular to the string counteracts the component of \(F_q\) along the string. This gives us two equations: \(m \cdot g \cdot \cos \theta = \frac{\sigma \cdot q}{2 \cdot \varepsilon_0} \cdot \sin \theta\) and \(m \cdot g \cdot \sin \theta = \frac{\sigma \cdot q}{2 \cdot \varepsilon_0} \cdot \cos \theta\)

Step 4: Solve for tan(theta)

Divide the first equation by the second, which yields \(\tan \theta = \frac{\sigma \cdot q}{2 \cdot \varepsilon_0 \cdot m \cdot g}\). This is the final result.