Dipole Moment: Study with Video Lessons, Practice Problems & Examples

An electric dipole consists of two equal but opposite charges separated by a distance, characterized by the dipole moment, represented as P = QD. When placed in a uniform electric field, the dipole gains potential energy, calculated using U = −PEcos(θ), and experiences torque, given by τ = PEsin(θ). Understanding these concepts is crucial for analyzing electric interactions and energy transformations.

Intro To Dipole Moment

Alright, guys. So for this video, we're going to be talking about the electric dipoles, and specifically there's one equation you need to know called the dipole moment. Let's go ahead and check it out. So whenever you have two equal charges but with opposite signs, like positives and negatives, and they're separated by some distance, they form what's called an electric dipole. So, for instance, you have a positive q and a negative q and they're separated by some distance d or r, whatever they form what's called an electric dipole. And the specific equation that you need to know is called the dipole moment. And that's this equation right here: P = Qd. The one thing you need to remember about this equation is that this is a vector. So, for instance, this P right here has a vector symbol on it, which means that when we represent it, we're going to need something in the I hat direction plus something in the J hat direction. That's how we represent vectors. Okay, so the q that you're going to use is just the magnitude of either of the charges. You don't have to pick which one's positive or negative; it's always just going to be the positive number. And because these things are going to be equal and opposite, you don't have to choose which one is the receiving charge or which one is the producing charge. This is just going to be the same. So, for instance, if I had like two coulombs and negative two coulombs, you would just put a two to infer that q. Okay, and this d here is a vector, and this vector points from the positive charge to the negative charge. Just how it always has been for electric field lines for parallel plate capacitors, things like that. This d is just a vector that points from the positive towards the negative. Okay, that's pretty straightforward. So let's go ahead and check out a quick example. So I asked to figure out what the vector dipole moment is of this following dipole right here. Now, the reason it says the vector dipole moment is that they actually wanted to represent it in vector form. So we need P = Qd. If they said the magnitude of the dipole moment, you would just have to do this without the actual vector form. Okay? So here's how you would do this. We need to figure out the vector dipole moment. So the charge is just two coulombs and negative two coulombs. So that's going to be the q that we replace in there. Now we just need the distance vector that always points from the positive towards the negative. So that's going to be over here in this direction. So this is my d vector. My d vector needs to be written in vector form. So, in other words, the X direction is the I hat direction. Remember that the Y direction is J hat. So how do we get from here all the way down to here? Well, we have to go negative 0.5 m in the I hat direction and then minus 1 m in the J hat direction. So this is our actual vector. Now all we do is just stick it inside this equation right here. So that means that our vector dipole moment is going to be two coulombs divided. And then we're going to distribute it inside of this vector right here. Negative 0.5 in the I hat direction minus one in the J hat direction. And let me wrap that up, and we get that the vector dipole moment, you just distribute this two inside. We're going to get negative one coulomb meter in the I hat direction and minus two coulomb meter in the J hat direction, and that's it. That's how you represent this vector dipole moments. Alright, let me know if you guys have any questions. Let's go ahead and take a look at another example.

Energy & Torque of Dipole Moments

All right, everyone, welcome back. So in the last video, we saw that an electric dipole has something called a dipole moment in between the two charges. Well, now we're going to see what happens if you put it in an electric field. Specifically, what happens to the energy? Let's check it out. So we put an electric dipole inside of a Uniform Electric field. It picks up a potential energy, and it's because basically the charges want to start moving around. And this potential energy right here is given by this fancy little equation, which is like a dot product between two vectors. But the more common expression that you'll see and the one that we'll be working with is this:

U = - p E cos θ

So the way I like to remember this is think back to when we're working in physics, one with a work where you have, like, a force off of in this direction. And if we had a distance in this direction, D then if we could figure out the angle between them theta, then the work was just equal to FD cos θ. It's kind of similar to that. We just need the components of the dipole moment that are in the same direction as the electric field. So we're going to use the cos of θ. And this torque, the reason that picks up a torque A dipole experiences a torque and electric field, is because if you think about the uniform electric field, let's say I had a uniform electric field pointing upwards like my fingers right here, and you had an electric dipole that was made up of a positive and a negative charge, and we knew that the dipole moment between them was P. Well, if you have P, which is the dipole moment and you have the electric field and what happens is we know positive charges, want to move upwards, and electric fields and negative charges want to move downwards. So what happens if you have this electric field right here? And this pen represents this dipole? The left charge wants to move up, and the right charge wants to move down, and we'll start to spin this whole entire thing. So that's called a torque. It creates like a rotational motion. And this is why dipoles experience torque inside of fields and So just like we worked with torques back in physics one, we just used the Cos, or sorry, the instead of the cosine, we used the sine of theta to describe the magnitude of that torque. And that's basically the two equations that you need to know. So again, let's go ahead and work out a very quick example.

So we have a dipole that's depicted in this figure below were told that there is a uniform electric field of 200 Newtons per Coulomb, and we're told to find what the potential energy is of the dipole. So in part A, we're going to figure out what the potential energy is. So we have two choices we could use p ⋅ E or we could use - p E cos θ . So we're going to use - p E cos θ because we have this angle right here that we're working with. So we know what the electric field is, and we just have to figure out what the angle between the dipole moment is and the electric field. So we actually be very careful about this. Remember that the electric dipole points from the distance vector. It points from the positive charge all the way to the negative charge. In other words, this vector right here is actually our dipole moment. So we need to figure out what the cosine or with the angle is between the electric field and the dipole moment. Now, thankfully, în this figure, this angle is 30 degrees. So we do know what the cosine of this angle is. So I just want to warn you because if this was the opposite, if this was actually like this, if this was actually backwards, and the dipole moment was like this, the angle would not be 30 degrees. It would be 150 degrees. So don't get that confused. So it's like this. So you just have to be careful, and then we just have to figure out what the actual dipole moment is, right? So we have to go figure out what that is. So the dipole moment is just going to be the magnitude is just going to be Q times D. Well, that dipole moment is just the charge involved, which is one Coulomb, times the distance. But remember, everything has to be in the right units. We have to have meters. Things like that, this is given as one centimeter as the distance vector between these two things. So that means that we're just going to use 0.01 in terms of meters. Okay, so we're just going to get 0.01 m. And now we can go ahead and stick it inside of the potential energy. So that means that our potential energy is just equal to negative we got 0.01 m we've got the electric field, which is 200 then we've got the cosine of theta. So the cosine of theta is the cosine of 30. And if you work this all out, you should get a potential energy of negative 1.73 joules.

Okay, so now let's look at part B. Part B is asking us to figure out what the torque is, and we're just going to use the p E sin θ , and that's it. That's all we're going to do. So we know what the torque is, and by the way, we're asked for the magnitude of the torque so with this, torque is usually given as a vector. But we can just figure out the magnitude by using this equation right here. We don't really have to worry about the direction. So the magnitude is just going to be p E sin θ and we already have all those numbers so let's go and plug them in. So the torque is just equal to 0.01 for the dipole moment, the electric field is 200 now the sin of 30 is going to give us our answer. So it means that the tau, the torque, is just going to be one, and that's going to be Newton meter. And that's basically our answers. So we got one Newton meter for the torque, and then we've got the potential energy of negative 1.73 joules. Don't worry about the fact that the potential energy is negative, because that just means that we're really only interested in the change in potential energy and versus kinetic energy. Alright, So don't don't get too preoccupied about the negative sign there. Alright, guys, let me know if you have any questions.

Here’s what students ask on this topic:

What is the dipole moment and how is it calculated?

The dipole moment is a measure of the separation of positive and negative charges in a system. It is calculated using the equation $P=QD$, where $Q$ is the magnitude of one of the charges and $D$ is the distance between the charges. The dipole moment is a vector quantity, pointing from the positive charge to the negative charge.

How does an electric dipole behave in a uniform electric field?

When an electric dipole is placed in a uniform electric field, it experiences both potential energy and torque. The potential energy $U$ is given by $U=-PEcos\left(\theta \right)$, where $P$ is the dipole moment, $E$ is the electric field strength, and $\theta$ is the angle between the dipole moment and the electric field. The torque $\tau$ is given by $\tau =PEsin\left(\theta \right)$, causing the dipole to align with the electric field.

What is the significance of the angle θ in the equations for potential energy and torque of a dipole in an electric field?

The angle $\theta$ represents the angle between the dipole moment vector $P$ and the electric field vector $E$. In the potential energy equation $U=-PEcos\left(\theta \right)$, $\theta$ determines how much of the dipole moment is aligned with the electric field. In the torque equation $\tau =PEsin\left(\theta \right)$, $\theta$ determines the rotational effect on the dipole, with maximum torque occurring when $\theta$ is 90 degrees.

How do you represent the dipole moment in vector form?

To represent the dipole moment in vector form, you need to consider both the magnitude and direction. The dipole moment vector $P$ is given by $P=QD$, where $Q$ is the charge and $D$ is the distance vector pointing from the positive to the negative charge. For example, if the distance vector is $(-0.5,-1)$ meters, and the charge is 2 Coulombs, the dipole moment vector would be $(-1,-2)$ Coulomb-meters.

What is the relationship between dipole moment and potential energy in an electric field?

The potential energy $U$ of a dipole in an electric field is given by the equation $U=-PEcos\left(\theta \right)$, where $P$ is the dipole moment, $E$ is the electric field strength, and $\theta$ is the angle between the dipole moment and the electric field. This relationship shows that the potential energy is minimized when the dipole is aligned with the electric field (θ = 0°) and maximized when it is anti-aligned (θ = 180°).