Review. An electric dipole in a uniform horizontal electric...

Key Concepts

Reduced Mass in Oscillatory Systems

When the two charges in a dipole have different masses, the dynamics of the system can be analyzed more easily using the concept of reduced mass. The reduced mass represents an effective inertial parameter of the two-body system that simplifies the calculation of the oscillation frequency in the presence of non-identical masses.

Small-Angle Approximation

In the context of oscillatory motion, particularly when the angular displacement is small, the small-angle approximation allows the sine of the angle to be approximated by the angle itself (in radians). This approximation simplifies the mathematical treatment of the problem, leading to linear differential equations characteristic of simple harmonic motion.

Moment of Inertia

The moment of inertia quantifies the rotational inertia of a rigid body and is a measure of how mass is distributed with respect to the axis of rotation. In the analysis of the oscillatory motion of an electric dipole, the moment of inertia of the system influences the angular acceleration produced by the restoring torque. It must be determined correctly to predict the oscillation frequency.

Simple Harmonic Motion in Rotational Systems

If a dipole is displaced slightly from its equilibrium orientation in a uniform electric field, the restoring torque acting on it is approximately proportional to the angular displacement, leading to simple harmonic motion. Analyzing this motion allows one to derive expressions for the oscillation frequency, which is a measure of how fast the dipole oscillates about the equilibrium position.

Torque in an Electric Field

When an electric dipole is placed in a uniform electric field, the field exerts forces on the two charges in opposite directions. These forces produce a torque that tends to align the dipole with the direction of the electric field. The torque is given by the cross product of the dipole moment and the electric field, and it plays a crucial role in determining the dipole's rotational dynamics.

Electric Dipole

An electric dipole consists of two equal but opposite charges separated by a fixed distance. This configuration creates an electric dipole moment, which characterizes the strength and direction of the dipole. The interaction of the dipole with external electric fields is fundamental in understanding phenomena like torque and potential energy in fields.

Transcript

00:01 So according to problem 15, there's an electric dipole, meaning a configuration of same and opposite charges, same and opposite charges.

00:13 Inside a uniform electric field, a vector.

00:18 It says horizontal meaning it points.

00:21 Let's take it that it points, yeah.

00:23 To the right.

00:24 Okay, the separation is between the two charges is two times a.

00:32 And it says that the electric dipole is displaced.

00:37 Maybe a better expression is that it is rotated, rotated about its center of mass, by an external factor or force anyway.

00:49 And at some point, this external factor disappears.

00:58 And naturally, the dipole tends to go back to its equilibrium position.

01:06 Okay.

01:07 Okay, so we have a restoring force as we will see in a minute.

01:11 And the exercise has two tasks.

01:15 It asks for the case that the two charges of the dipole have exactly the same mass to calculate the frequency of oscillation and that in that this is the frequency of oscillation.

01:34 And the that the two charges have a different mass with each other that the frequency of oscillation changes to this expression here.

01:45 Okay, let's continue immediately.

01:48 So, this kind of exercises basically is exactly what we do.

01:56 This kind of exercises follow exactly what we do in the case that we need to calculate the frequency of oscillation for harmonic oscillations.

02:06 Let's do it, let's first draw the forces accepted on the two charges.

02:14 So that's an electric point in to the right.

02:17 Therefore, for the positive charge, there's an lp force which points again to the right.

02:24 But for the negative recharge particle, then naturally there's an electric force pointing to the left.

02:30 Okay, but the magnitude of all the two forces is exactly the same.

02:37 And let's use e for the magnitude of the electric field.

02:41 Okay, and the magnitude of the force is just two times e.

02:45 Okay, now, let's, let's, let's, let's say, is the force accepted on the two charges that forces, forces that we charge us to stay to stay as a part of the dipole.

03:07 Okay.

03:09 Now if we decompose t, this constant force there, into two components, one component that is parallel to the electric field, another one that is perpendicular to the electric field.

03:22 Again, the simple indicates parallel, this symbol indicates perpendicular.

03:27 Okay, we can say it is quite easy to see exactly as we usually do in exercises with harmonic oscillator that the parallel component is cancelled by the electric force therefore the magnitude of our t parallel which equals a simple trichonometry this is t times cosine theta equals q e in the electric force and now the other component t perpendicular which is the restoring forces the force that pushes the charge uh yeah except on the on the charges support the charges and this is the force that forces the the dipole to go back to its equilibrium position okay that's why we call it restoring force and again simple trigonometry if this is theta the magnet of t perpendicular is just t times sine theta and if we place t from here from this equation meaning t equals qe divided by cost theta which is divided divided by cost theta times sine theta this expression is simply the definition of tangent theta.

05:05 And now, if we take theta small, it's been given by the guidelines.

05:13 Okay.

05:14 Then we end up with an expression for the magnitude of the restoring force as q times a times, times theta.

05:20 Okay.

05:22 Now, now that we have the magnitude of the restoring force, we can work out to find out the frequency of oscillation.

05:32 Okay.

05:35 Again, it's the same...