$U=-B\cdot \mu$ is defined to be the magnetic potential energy, I saw this in my lecture notes, but we had already talked about the fact that since the work done to move a charge there is path dependant, there can not be a unique potential at a point in a magnetic field. So there is no use in defining a magnetic potential energy.

The only thing I have in my notes is that it shows for a current loop, it has the most potential energy when the magnetic moment is perpendicular to the field, and has 0 potential energy when it is parallel. I can sort of justify this by saying it makes sense that the potential should be a maximum in an orientation where there is the most restoring torque on the loop and minimum where there is none

But what does a potential mean in a non conservative field?


1 Answer 1


The problem you're having is that your confusing a field's scalar "potential" with potential energy. The concepts are related, but in a particular way. Say I have a vector field $\mathbf{V}(\mathbf{x})$. There is a mathematical theorem that I can break this field down into two two parts $$\mathbf{V}(\mathbf{x}) = -\nabla \Phi + \nabla\times\mathbf{A},$$ $\Phi$ is something we call $\mathbf{V}$'s scalar potential and $\mathbf{A}$ is its vector potential.

The connection to potential energy comes in if we can write a force on some particle is of the form $$\mathbf{F} = a \mathbf{V}$$ and $\mathbf{A} = 0$, then the potential energy of that particle will be $U = a\Phi$.

So, in the case you're dealing with you don't have a force of needed form, so it's not a problem. Granted, we still have the statement, "The magnetic field can do no work," but showing how that works out in the case of a magnetic dipole like this is subtle. In fact, the force and torque on a magnetic dipole are \begin{align} \mathbf{F} & = \left(\mathbf{\mu}\cdot\nabla\right)\mathbf{B} \text{ and} \\ \mathbf{\tau} & = \mathbf{\mu}\times \mathbf{B}. \end{align} I'm less certain here, but I'm pretty sure you'll find that the combination of those two makes it impossible to harvest infinite energy from any static magnetic field.

  • $\begingroup$ Are you saying the electric field set up by the current loop does the work, and the magnetic potential energy is just a store of the electric fields KE? I dont think I understand your answer, sorry can you please clarify. $\endgroup$ May 24, 2019 at 7:59
  • $\begingroup$ @VishalJain Completely reworked answer. It's possible to make what was said in the old answer relevant, but it requires a bit of work. $\endgroup$ May 24, 2019 at 8:26
  • $\begingroup$ To summarise, you are saying since the vector potential is non zero, the potential energy is not just the scalar potential, which is what gravitational and electric potential is. I understand that my definition of potential energy is the issue, but if the potential energy isnt just the scalar potential, what physical significance does the vector potential have. I understand the scalar potential as the work to move a charge from infinity to a point, what analog can be used for the vector potential? $\endgroup$ May 24, 2019 at 8:35
  • $\begingroup$ The magnetic vector potential is difficult to interpret. See, you recall how we can shift the scalar potential by a constant value and no physics changes? Well, it turns out that's a smaller part of something called a "gauge transform" that makes a large chunk of $\mathbf{A}$ arbitrary. Point being, that's a separate question, not a comment on an answer. $\endgroup$ May 24, 2019 at 8:49
  • $\begingroup$ ill cross that bridge when i get to it, thanks $\endgroup$ May 24, 2019 at 9:11

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