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I understand that the torque acting on a magnetic dipole in a magnetic field is given by $$\vec\tau=\vec\mu\times\vec B \therefore \tau = \mu B sin\phi$$ It follows that the work done as the dipole moves through an infinitesimal angular displacement is $$dW=\tau d \phi \therefore W=\mu B\int sin\phi d\phi = -\mu Bcos\phi \therefore W=-\vec\mu\cdot \vec B$$ Where I have become confused is that the potential energy of a magnetic dipole in a magnetic field is given by the same expression: $U=-\vec\mu\cdot\vec B $. So, if the work done on a magnetic dipole through an angular displacement is given by the negative expression above, shouldn't the expression for potential energy then be: $U=\vec\mu\cdot\vec B$ rather than the negative of the same expression?

The only thing I can think of is if the torque is caused by an external force then the change in potential energy is equal to the work done by the external torque but I have a feeling this is not the correct way of looking at it.

Any help would be greatly appreciated!

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The negative sign indicates the tendency of the dipole to be aligned with the magnetic field; a dipole transverse to its local magnetic field has more energy than when it's parallel to the field.

Looking at your computation, it looks like the confusion lies in how you've defined $\theta$. In the torque expression: $\tau = \mu B\sin\theta$, $\theta$ is defined to be the angle between the moment and the magnetic field. With this definition, $\text{d}\theta$ corresponds to a rotation of the dipole away from the direction of the magnetic field.

Now your work expression: $\text{d}W = \tau\text{d}\theta$, says that work is done by the magnetic field when the dipole rotates away from the axis of the magnetic field, which is incorrect (the magnetic field works in the opposite way). Hence, you should write $\text{d}W = -\tau \text{d}\theta$, which gives the correct answer.

Edit: actually there's an easier way to see this. The torque on a moment by the magnetic field is a restoring force. That is, the magnetic field wants to rotate the moment towards $\theta = 0$. Hence, you should write $\tau = -\mu B \sin\theta$ and proceed with your original calculation. Either way, you need to use the fact that the dipole has a tendency to align with the magnetic field.

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  • $\begingroup$ Thank you, this really cleared up a lot of confusion for me! I think that I understand what you're saying about torque being a restoring force, however, what causes the angular displacement in the first place? Or is it not important? Sorry if that seems like a silly question but I think where I am getting confused still is I keep thinking that the torque is what causes the angular displacement. Thanks! $\endgroup$
    – Thomas
    Commented May 27, 2021 at 14:22
  • $\begingroup$ No problem! You are correct in that the torque due to the magnetic field is what causes the angular displacement. In the "restoring force" narrative, the torque $\tau = -\mu B\sin\theta$ causes a displacement $\text{d}\theta$, which does work $$\text{d}W = \tau\text{d}\theta = -\mu B\sin\theta\text{d}\theta.$$ $\endgroup$
    – jsborne
    Commented May 27, 2021 at 14:27
  • $\begingroup$ So would it be correct to say that the angle $\theta$ between $\vec\mu$ and $\vec B$ is the initial position and the torque does work over that angle but in the opposite direction in order to restore it to $\theta = 0$, if that makes sense? $\endgroup$
    – Thomas
    Commented May 27, 2021 at 15:43
  • $\begingroup$ That's exactly right. $\endgroup$
    – jsborne
    Commented May 27, 2021 at 16:40

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