Question Video: Defining the Magnetic Dipole Moment Physics

Video Transcript

Which of the following formulas correctly describes the relation of 𝑚 sub 𝑑, the magnetic dipole moment of a loop of current-carrying wire in a uniform magnetic field, to 𝜏, the torque acting on the loop, and 𝐵, the magnitude of the magnetic field? (A) 𝑚 sub 𝑑 equals 𝜏 over 𝐵. (B) 𝑚 sub 𝑑 equals 𝐵𝜏. (C) 𝑚 sub 𝑑 equals 𝐵 over 𝜏. (D) 𝑚 sub 𝑑 equals 𝐵 plus 𝜏. (E) 𝑚 sub 𝑑 equals 𝐵𝜏 over 𝐵 plus 𝜏.

Here, we’re asked to consider a loop of current-carrying wire that is within a uniform magnetic field and figure out which of the given formulas correctly describe the relation of the magnetic dipole moment, 𝑚 sub 𝑑, the torque acting on the loop, 𝜏, and the magnitude of the magnetic field, 𝐵. Let’s begin by reminding ourselves what happens when we have a loop of current-carrying wire that resides in a uniform magnetic field and what the magnetic dipole moment of a loop of current-carrying wire would be.

Recall that when we have a loop of wire that is carrying some electric current that is within a magnetic field, a force acts upon it. We call the force acting on a loop a torque, which causes it to spin within the magnetic field. To describe the torque acting on a loop of wire, we use the equation torque, 𝜏, is equal to the magnitude of the magnetic field the loop is in, 𝐵, multiplied by the current that flows through the loop, 𝐼, multiplied by the area of the loop, 𝐴. We have the equation for torque now, which is good since it is present in all of the answer choices. So now, we need to figure out the magnetic dipole moment.

We can start by taking the definition of the magnetic dipole moment of a loop of current-carrying wire. The magnetic dipole moment, 𝑚 sub 𝑑, is equal to the current through the loop, 𝐼, multiplied by the area of the loop, 𝐴. Notice that the equation for the torque acting on the current-carrying loop also has the values of the current, 𝐼, multiplied by the area, 𝐴. This means that we can replace those values in our equation for torque with the magnetic moment.

We now have the equation torque acting on the loop, 𝜏, is equal to the magnitude of the magnetic field, 𝐵, multiplied by the magnetic dipole moment, 𝑚 sub 𝑑. We can now rearrange this equation to get an equation in terms of the magnetic dipole moment. We can do this by dividing both sides of the equation by the magnitude of the magnetic field, 𝐵. We can cancel the 𝐵 divided by 𝐵 on the right-hand side, and we will be left with our final equation. The magnetic dipole moment of a current-carrying loop in a uniform magnetic field, 𝑚 sub 𝑑, is equal to the torque acting on the loop, 𝜏, divided by the magnitude of the magnetic field it is in, 𝐵.

If we compare this equation with our answer options, we can see that this equation matches option (A), 𝑚 sub 𝑑 equals 𝜏 over 𝐵. So, this must be the correct answer.