An electric dipole in a uniform horizontal electric field is...

An electric dipole in a uniform horizontal electric field is displaced slightly from its equilibrium position as shown in Figure $\mathrm{P} 23.79,$ where $\theta$ is small. The sepa- ration of the charges is $2 a,$ and each of the two particles has mass $m$ (a) Assuming the dipole is released from this position, show that its angular orientation exhibits simple harmonic motion with a frequency $$f=\frac{1}{2 \pi} \sqrt{\frac{q E}{m a}}$$ What If? (b) Suppose the masses of the two charged particles in the dipole are not the same even though each particle continues to have charge $q$ . Let the masses of the particles be $m_{1}$ and $m_{2}$ . Show that the frequency of the oscillation in this case is $$f=\frac{1}{2 \pi} \sqrt{\frac{q E\left(m_{1}+m_{2}\right)}{2 a m_{1} m_{2}}}$$

An electric dipole in a uniform horizontal electric field is displaced slightly from its equilibrium position as shown in Figure $\mathrm{P} 23.79,$ where $\theta$ is small. The sepa- ration of the charges is $2 a,$ and each of the two particles has mass $m$ (a) Assuming the dipole is released from this position, show that its angular orientation exhibits simple harmonic motion with a frequency
$$f=\frac{1}{2 \pi} \sqrt{\frac{q E}{m a}}$$
What If? (b) Suppose the masses of the two charged particles in the dipole are not the same even though each particle continues to have charge $q$ . Let the masses of the particles be $m_{1}$ and $m_{2}$ . Show that the frequency of the oscillation in this case is
$$f=\frac{1}{2 \pi} \sqrt{\frac{q E\left(m_{1}+m_{2}\right)}{2 a m_{1} m_{2}}}$$

Key Concepts

Moment of Inertia

The moment of inertia quantifies the resistance of a body to angular acceleration about a given axis, depending on the mass distribution relative to that axis. In the case of a dipole, the moment of inertia depends on the masses of the charged particles and their positions relative to the center of rotation. Understanding how to compute this rotational inertia is essential for linking the applied torque to the angular acceleration in the analysis of SHM.

Simple Harmonic Motion (SHM)

Simple harmonic motion describes a type of periodic oscillation where the restoring force is directly proportional to the displacement from equilibrium. In the context of a dipole, the small-angle approximation allows the restoring torque to be expressed as proportional to the angle, thereby leading to angular SHM. The analysis of SHM provides a framework for determining the oscillation frequency of the dipole based on system parameters.

Small-Angle Approximation

For small angular displacements, the sine of the angle is approximately equal to the angle itself (measured in radians). This approximation simplifies the equations of motion for the rotating dipole, transforming the torque equation into one analogous to that governing simple harmonic motion. It is a key mathematical tool that enables the derivation of an expression for the oscillatory frequency.

Uniform Electric Field

A uniform electric field has constant magnitude and direction at every point in space. When a dipole is placed in such a field, the forces on the individual charges are equal in magnitude and opposite in direction, resulting in a torque that tends to align the dipole with the field. This simplification allows for the analysis of the dipole’s rotational dynamics without the complication of net translational forces.

Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a distance. Its behavior in external fields is characterized by its dipole moment, a vector that describes the magnitude and direction of the charge separation. This concept is central to understanding interactions with electric fields, as the dipole experiences both a net force (if the field is non-uniform) and a torque (even in a uniform field).

Torque and Restoring Torque

When a dipole is displaced from equilibrium in an electric field, the forces on the dipole charges produce a torque that acts to restore the dipole to alignment with the field. This restoring torque, which is proportional to the sine of the displacement angle for small angles, is fundamental to setting up the dynamics of oscillatory motion. The concept bridges rotational dynamics with the principles of oscillator motion.

Transcript

00:01 So the net torque can be equal to the perpendicular distance times the force, which is equal to rf sine theta.

00:08 Invention, we have negative 2a sine theta times qe.

00:11 Okay? and the torque can be also equal to the moment of inertia times the second derivative of charge over time.

00:19 And this will give us negative omega square, which is angular speed square, times the angle theta, and then times the moment of inertia.

00:35 So therefore, we have negative omega square theta times moment of inertia is equal to negative 2a, sine theta, times qe.

00:59 Okay? it's because the electric force on each charge is along the horizontal direction.

01:08 So therefore, the perpendicular distance that is relative to the electric force must be the vertical direction, which is the perpendicular distance, okay, which is 2a sine theta.

01:21 And then we know that sine theta can be approximately equal to theta, since the angle theta is really small in this case, okay? so therefore, we can substitute back into the equation, then we have negative omega -square -theta times i is equal to negative 2a -theta times q.

01:50 Then as you can tell, negative sign can be cancelled out and theta, theta can be cancelled out.

01:57 Therefore we have omega square times i is equal to 2aqe.

02:10 Then we know angular velocity can be equal to 2 pi times frequency.

02:19 Therefore we'll have 2 pi times f square is equal to 2qae over moment of inertia...

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