What are the correct units for the magnetic dipole moment of a coil?
Now to answer this question, let’s first think about what the words magnetic dipole moment actually mean. And to understand this, let’s not think about a coil just yet. Let’s first think about a bar magnet. So here’s our bar magnet. And we’ve drawn it in such a way that the north pole is on the right and the south pole is on the left. Now, let’s assume we placed this bar magnet in an external magnetic field. And let’s also say that this magnetic field is pointing upwards.
Well, in this situation, our little bar magnet is going to experience a torque. Specifically, there are going to be forces on the ends of the bar magnet which end up rotating it so that it’s aligned with the external magnetic field B. And the bar magnet’s magnetic dipole moment is closely related to this. We can recall that the torque experienced by our little bar magnet is defined as the magnetic dipole moment — which we’ll call 𝜇 — multiplied by the strength of the magnetic field B. Or in other words, we can rearrange the equation to solve for 𝜇. And we can see that the magnetic dipole moment is equal to the torque per unit magnetic field experienced by the bar magnet.
Essentially, what this means is that if we place a bar magnet with a magnetic moment 𝜇 inside a specific magnetic field, we can work out how much torque is going to experience in that magnetic field. And we can use this equation to work out the units of the magnetic dipole moment because these units are gonna be equal to the units of torque divided by the units of magnetic field. So let’s first recall what torque means.
Torque is defined as a force multiplied by the perpendicular distance between the point at which the force is applied and the point at which an object is rotating. So in our scenario, we’ve got a force 𝐹 applied to the end of a bar magnet. And of course, we’ve got another force 𝐹 here as well. But for now, let’s just consider this one. And that force 𝐹 is causing our bar magnet to rotate about the center of the magnet. So to find the torque exerted by this force here, we simply multiply the magnitude of that force by the distance between the point at which the force acts and the point at which the object is rotating, so this distance here. And we can do the same for this force over here as well. But the important thing is that torque is defined as a force multiplied by a distance.
And therefore, if we want to work out the units of torque, that’s what the square brackets around the 𝜏 mean by the way we’re talking about the units, then this is going to be the unit of force — that’s newtons — multiplied by the unit of distance — that’s meters. So we now know that torque has units of newton meters. And we can also recall that the magnetic field or magnetic field strength or magnetic flux density — whichever we wanna call it — has units of tesla, which means that we can now work out the units of magnetic dipole moment. That is going to be equal to the unit of torque divided by the unit of magnetic field strength. And when we substitute in these units here, we get newton meters per tesla.
And so in general, magnets have a magnetic dipole moment. And the unit of magnetic dipole moment is newton meters per tesla. So this is one potential answer for our question, even though we haven’t considered a coil. But actually, we could consider a coil and get a very slightly different answer. Let’s now think about a current loop.
We’ve got basically a circular piece of wire with a current 𝐼 flowing through it, anticlockwise as we’ve drawn it. Well, in this scenario, we can recall that a loop of current can act like a magnetic dipole. In other words, it basically behaves like a little bar magnet. We can even draw in some more magnetic field lines to show that they would match those of a bar magnet. And so, if a current loop acts like a bar magnet, then it must have a magnetic dipole moment as well. And specifically, for loops of current, the magnetic dipole moment 𝜇 ends up being equal to the current flowing through that loop multiplied by the area of that loop. We’ll call that area 𝐴.
Now happily for us, we’ve considered the current loop which also is the same thing as a coil because a coil is simply a loop of current. Remember that the coil doesn’t have to be circular. It could even be rectangular like we see in a dynamo for example or lots of different other shapes as well. However, the magnetic dipole moment of a coil is always given by the current flow through that coil multiplied by its area. So if that’s the definition of the magnetic moment of a coil, then the units of the magnetic dipole moment are going to be equal to the units of current multiplied by the units of area. And we can recall that the unit of current is the ampere and the standard unit of area is meters squared, which means that we’ve come up with another answer for the units of magnetic dipole moment of a coil. The units are amp meter squared.
Now, it’s important to know that although these two answers look different, the two sets of units are in fact equivalent to each other. And so, we can choose to give either one of these as the answer to our question. Therefore, the correct units for the magnetic dipole moment of a coil are either newton meters per tesla or amp meters squared or any other equivalent correct set of units.