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Chapter 23: Problem 55

Frequency of Oscillation Find the frequency of oscillation of an electric dipole, of dipole moment \(\vec{p}\) and rotational inertia \(I\), for small amplitudes of oscillation about its equilibrium position in a uniform electric field of magnitude \(|\vec{E}|\)

Short Answer

Expert verified
The frequency is \( f = \frac{1}{2\pi} \sqrt{ \frac{pE}{I} } \).

Step by step solution

01

Identify the torque

For an electric dipole in a uniform electric field, the torque \(\tau\) is given by \(\tau = \vec{p} \times \vec{E}\). For small angular displacements \(\theta\), \( \tau \approx -pE\sin\theta \approx -pE \theta \).
02

Apply Newton's Second Law for Rotation

Using Newton's second law for rotation, \( I\alpha = \tau \, where \ alpha \) is the angular acceleration \( \frac{d^2 \theta}{dt^2}\). This gives \( I \frac{d^2 \theta}{dt^2} = -pE \theta \).
03

Formulate the Differential Equation

Rearrange the equation to \(\frac{d^2 \theta}{dt^2} + \frac{pE}{I} \theta = 0\). This is the standard form of a simple harmonic oscillator.
04

Identify the Angular Frequency

The standard form \( \frac{d^2 \theta}{dt^2} + \omega^2 \theta = 0 \) gives the angular frequency \( \omega = \sqrt{ \frac{pE}{I} } \).
05

Calculate the Frequency

The frequency of oscillation is \( f = \frac{\omega}{2\pi} = \frac{1}{2\pi} \sqrt{ \frac{pE}{I} } \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electric Dipole
An electric dipole consists of two equal and opposite charges, separated by a small distance. The dipole moment \( \vec{p} \ \) is a vector quantity pointing from the negative charge to the positive charge. The magnitude of the dipole moment is given by \( p = q \cdot d \ \), where \( q \ \) is the magnitude of the charge and \( d \ \) is the separation distance.
When placed in a uniform electric field \( \vec{E} \ \), the dipole experiences a torque \( \tau \ \) that tends to align it with the field. The torque is calculated as \( \tau = \vec{p} \times \vec{E} \ \). For small angles, we can approximate \( \sin\theta \ \) to \( \theta \ \), giving \( \tau \approx -pE\theta \ \).
Angular Frequency
Angular frequency \( \omega \ \) represents how fast something oscillates or rotates. For a simple harmonic oscillator, it is linked to the spring constant and mass, but in this scenario, it's related to rotational inertia \( I \ \) and dipole properties.
The formula \( \omega = \sqrt{ \frac{pE}{I} } \ \) shows that \( \omega \ \) depends on the dipole moment \( p \ \) and the electric field \( E \ \), indicating how quickly the dipole moves back and forth within an electric field.
This is similar to how a pendulum's angular frequency depends on gravity and length. Angular frequency is used to find the actual frequency \( f \ \) using \( f = \frac{\omega}{2\pi} \ \).
Simple Harmonic Oscillator
A simple harmonic oscillator describes motion where the force restoring the system to equilibrium is directly proportional to displacement. This leads to motion given by \( x(t) = A \cos(\omega t + \phi) \ \).
In our dipole's context, the restoring torque \( \tau = -pE\theta \ \) acts like a spring force, making the dipole swing back and forth.
The differential equation \( \frac{d^2\theta}{dt^2} + \frac{pE}{I}\theta = 0 \ \) mirrors the simple harmonic oscillator form \( \frac{d^2\theta}{dt^2} + \omega^2\theta = 0 \ \). This equation tells us that the motion is periodic and can be described by a sinusoidal function over time.
Newton's Second Law for Rotation
Newton's Second Law for Rotation states that the net torque \( \tau \ \) on a body is equal to the product of its moment of inertia \( I \ \) and its angular acceleration \( \alpha \ \), or \( \tau = I\alpha \ \).
In the dipole example, taking torque \( \tau = -pE\theta \ \) and substituting into Newton's Second Law gives us \( I \frac{d^2\theta}{dt^2} = -pE\theta \ \). Solving this yields the same simple harmonic oscillator equation.
This blend of rotational dynamics and harmonic motion explains the periodic swing of the dipole in an electric field, reflecting the core principles linking rotation and oscillation.

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Most popular questions from this chapter

What Distance? At what distance along the central axis of a ring of radius \(R\) and uniform charge is the magnitude of the electric field due to the ring's charge maximum?

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