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, if the string makes an angle \( \theta \) with the vertical, then
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, if the string makes an angle \( \theta \) with the vertical, then
Show Hint
For any particle under gravity and a constant horizontal force \( F_h \), the angle \( \theta \) of suspension with the vertical at equilibrium always satisfies \( \tan\theta = \frac{F_h}{mg} \). Simply substituting the electric force as \( F_h \) gives the result immediately.
Updated On: May 28, 2026
- \( \tan\theta = \frac{\sigma q}{2\varepsilon_0 mg} \)
- \( \tan\theta = \frac{\sigma q}{\varepsilon_0 mg} \)
- \( \cot\theta = \frac{\sigma q}{2\varepsilon_0 mg} \)
- \( \cot\theta = \frac{\sigma q}{\varepsilon_0 mg} \)
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Understanding the Question:
A simple pendulum with a charged bob is in equilibrium under the influence of gravity and a horizontal electrostatic force produced by an infinite vertical sheet of charge passing through its point of suspension. We need to find the relation for \( \tan\theta \), where \( \theta \) is the angle the string makes with the vertical at equilibrium.
Step 2: Key Formula or Approach:
1. Electric field of a sheet of charge: An infinite plane sheet of charge with surface charge density \( \sigma \) produces a uniform electric field directed perpendicular to the sheet:
\[ E = \frac{\sigma}{2\varepsilon_0} \]
Since the sheet is vertical, the electric field is horizontal.
2. Electrostatic Force: The force on the charge \( q \) is:
\[ F_e = q E = \frac{\sigma q}{2\varepsilon_0} \]
3. Equilibrium of the bob: By resolving the tension \( T \) in the string of the pendulum into horizontal and vertical components:
\[ T \sin\theta = F_e \]
\[ T \cos\theta = mg \]
Step 3: Detailed Explanation:
Dividing the horizontal equilibrium equation by the vertical equilibrium equation:
\[ \frac{T \sin\theta}{T \cos\theta} = \frac{F_e}{mg} \]
\[ \tan\theta = \frac{\left( \frac{\sigma q}{2\varepsilon_0} \right)}{mg} \]
\[ \tan\theta = \frac{\sigma q}{2\varepsilon_0 mg} \]
Step 4: Final Answer:
The equilibrium angle \( \theta \) satisfies \( \tan\theta = \frac{\sigma q}{2\varepsilon_0 mg} \), which corresponds to option (A).
A simple pendulum with a charged bob is in equilibrium under the influence of gravity and a horizontal electrostatic force produced by an infinite vertical sheet of charge passing through its point of suspension. We need to find the relation for \( \tan\theta \), where \( \theta \) is the angle the string makes with the vertical at equilibrium.
Step 2: Key Formula or Approach:
1. Electric field of a sheet of charge: An infinite plane sheet of charge with surface charge density \( \sigma \) produces a uniform electric field directed perpendicular to the sheet:
\[ E = \frac{\sigma}{2\varepsilon_0} \]
Since the sheet is vertical, the electric field is horizontal.
2. Electrostatic Force: The force on the charge \( q \) is:
\[ F_e = q E = \frac{\sigma q}{2\varepsilon_0} \]
3. Equilibrium of the bob: By resolving the tension \( T \) in the string of the pendulum into horizontal and vertical components:
\[ T \sin\theta = F_e \]
\[ T \cos\theta = mg \]
Step 3: Detailed Explanation:
Dividing the horizontal equilibrium equation by the vertical equilibrium equation:
\[ \frac{T \sin\theta}{T \cos\theta} = \frac{F_e}{mg} \]
\[ \tan\theta = \frac{\left( \frac{\sigma q}{2\varepsilon_0} \right)}{mg} \]
\[ \tan\theta = \frac{\sigma q}{2\varepsilon_0 mg} \]
Step 4: Final Answer:
The equilibrium angle \( \theta \) satisfies \( \tan\theta = \frac{\sigma q}{2\varepsilon_0 mg} \), which corresponds to option (A).
Was this answer helpful?
0
0
Top WBJEE Physics Questions
- The velocity of a particle executing a simple harmonic motion is $13\, ms^{-1}$, when its distance from the equilibrium position (Q) is 3 m and its velocity is $12 \, ms^{-1}$, when it is 5 m away from The frequency of the simple harmonic motion is
- A and B are two parallel sided transparent slabs of refractive indices n$_1 $and n$_2$ respectively. A ray is incident at an angle $\theta$ on the surface of separation of A and B, and after refraction from B into air grazes the surface of B. Then
- The minimum and maximum capacitances, which may be obtained by the combination of three capacitors each of capacitance 6$\mu$ F are
- The increase in electrostatic potential energy of a dipole of moment p when it is taken from parallel to anti-parallel orientation in an electric field E is
- A zener diode has break down voltage of 5.0 V. The resistance required to allow a current of 100 mA through the zener in reverse bias when connected to a battery of emf 12 V is
View More Questions
Top WBJEE Pendulums Questions
- A simple pendulum is taken at a place where its distance from the Earth's surface is equal to the radius of the Earth. Calculate the time period of small oscillations if the length of the string is 4.0 m. (Take \( g = 9 \, \text{m/s}^2 \) at the surface of the Earth.)
- A simple pendulum is taken at a place where its distance from the earth’s surface is equal to the radius of the earth. Calculate the time period of small oscillations if the length is 4.0 m. (Take \( g=\pi^2\,\text{m/s}^2 \) at surface.)
Top WBJEE Questions
- Let $\vec{\alpha } = \hat{i} + \hat{j} + \hat{k} , \vec{\beta} = \hat{i} - \hat{j} - \hat{k}$ and $\vec{\gamma} = - \hat{i} + \hat{j} - \hat{k}$ be three vectors. A vector $\vec{\delta} $, in the plane of $\vec{\alpha}$ and $\vec{\beta}$, whose projection on $\vec{\gamma}$ is $\frac{1}{\sqrt{3}}$, is given by
- Let a, b, c be real numbers such that $a + b + c < 0$ and the quadratic equation $ax^2 + bx + c = 0$ has imaginary roots. Then
- If the radius of a spherical balloon increases by 0.1%, then its volume increases approximately by
- The velocity of a particle executing a simple harmonic motion is $13\, ms^{-1}$, when its distance from the equilibrium position (Q) is 3 m and its velocity is $12 \, ms^{-1}$, when it is 5 m away from The frequency of the simple harmonic motion is
- By which of the following methods, new and better varieties of plants can be formed?
View More Questions