An electric dipole with a dipole moment of magnitude p is...

An electric dipole with a dipole moment of magnitude $p$ is placed at various orientations in an electric field $\vec{E}$ that is directed to the left. (a) What orientation of the dipole will result in maximum torque directed into the page? What then is the electric potential energy? (b) What orientation of the dipole will give zero torque and maximum electric potential energy? What type of equilibrium is this: stable, unstable, or neutral?

An electric dipole with a dipole moment of magnitude $p$ is placed at various orientations in an electric field $\vec{E}$ that is directed to the left. (a) What orientation of the dipole will result in maximum torque directed into the page? What then is the electric potential energy?
(b) What orientation of the dipole will give zero torque and maximum electric potential energy? What type of equilibrium is this: stable, unstable, or neutral?

Key Concepts

Equilibrium Conditions

Equilibrium for a dipole in an electric field occurs when the net torque is zero, meaning the dipole is oriented along or opposite to the electric field. A stable equilibrium occurs when a slight disturbance results in forces that return the dipole to its original alignment, whereas an unstable equilibrium is characterized by any small disturbance causing further deviation from that configuration.

Electric Potential Energy of a Dipole

The electric potential energy of a dipole in an external electric field is determined by the dot product of the dipole moment and the electric field. This energy is lowest (most negative) when the dipole is aligned with the field and highest when it is anti-aligned with the field, indicating the stability of the aligned configuration.

Electric Dipole

An electric dipole consists of two equal but opposite charges separated by a distance, and it is characterized by its dipole moment—a vector that points from the negative to the positive charge. This dipole moment quantifies the dipole's overall effect in an electric field.

Torque on an Electric Dipole

When placed in an electric field, an electric dipole experiences a torque that tends to align the dipole with the field. The magnitude of this torque is given by the cross product of the dipole moment and the electric field. The torque reaches its maximum value when the dipole moment is perpendicular to the electric field.

Transcript

00:01 Hello, irma.

00:01 In this problem, we're asked to find various orientations of an electric dipole inside a constant electric field that is pointing to the left.

00:11 So the first part of the question is asking us, what orientation of the magnetic dipole is going to give us the maximum possible torque that is pointing into the page? so here what we have is that, you know, the dipole moment is defined as the vector connecting the negative to the positive charge, as i've demonstrated over here.

00:35 And the torque given on the dipole is this dipole moment cross -productive or cross -multiplied with the electric field e.

00:43 So what this means is that we're going to use a right -hand rule where our fingers, index fingers, and all the other fingers, except for the thumb, are pointing along the direction of p.

00:53 And we're going to curl our fingers towards e to get the torque.

00:58 So you can convince yourself that if you have that the orientation is such that p is pointing down, so if you have the negative charge up there and the positive church down here, then you're going to draw p as connecting the two of them such that p is pointing directly down.

01:17 So in this case, there's no angle.

01:19 If you were to write this out in terms of, you know, what is the magnitude of the torque? so this is this is the vector.

01:28 But the magnitude of it is going to be p times e times sign of some angle.

01:35 Let's say phi, that is the angle between the two vectors.

01:39 So obviously you're going to have a maximum torque when phi is either 90 degrees or a multiple of 90 degrees.

01:47 Right? so pi over two, three pi over two, et cetera.

01:51 So in this case, we have that the angle between the two vectors, so the electric field and the dipole moment is 90 degrees.

02:03 So we're going to have a maximum torque.

02:06 But which way is this torque pointing? well, so now if we look at this, then this will want to move.

02:15 If you just kind of look at what the forces do, then the electric field is going to to push the bottom charge, the positive charge, to the left.

02:22 So that's going to be f on the positive charge.

02:27 And the negative charge is going to be attracted to the right by this electric field.

02:32 So there's going to be an f of a coolant force that's attracting the negative charge to the right.

02:38 And so with that, you can see that the torque is now going to be counterclockwise, the way we'll look at it now.

02:45 And so it's going to be pointing into the page.

02:47 And because it's at a 90 degree angle with respect to the electric field, so it's perpendicular to it, the torque is in fact going to be maximum.

02:55 So this is the correct orientation for maximum torque...

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