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KDPhysics

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Recently I have encountered the following expression for the potential energy of a magnetic dipole of moment ##\boldsymbol{\mu}## placed in an external magnetostatic field

$$U=-\boldsymbol{\mu} \cdot \textbf{B}$$.

However, I was told that magnetic fields are non-conservative, so we can't define a scalar potential and thus potential energy.

Since magnetic fields do no work, how can we have a magnetic potential energy? Is it the magnetic torque that does work?

In such case, then I find that given that the magnetic torque is ##\boldsymbol{\tau} =\boldsymbol{\mu} \times \textbf{B}## and hence the work done is:

$$W=-\int_{\frac{\pi}{2}}^\theta \tau d\theta = - \int_{\frac{\pi}{2}}^\theta \mu B \sin \theta d\theta = \mu B \cos \theta = \boldsymbol{\mu} \cdot \textbf{B}$$

where ##\theta## is the angle between the magnetic moment and the external magnetic field. Thus the potential energy is ##-\boldsymbol{\mu} \cdot \textbf{B}## as required.

But it still isn't clear, what causes this potential energy to exist? Magnetic forces do no work after all.

**B:**$$U=-\boldsymbol{\mu} \cdot \textbf{B}$$.

However, I was told that magnetic fields are non-conservative, so we can't define a scalar potential and thus potential energy.

Since magnetic fields do no work, how can we have a magnetic potential energy? Is it the magnetic torque that does work?

In such case, then I find that given that the magnetic torque is ##\boldsymbol{\tau} =\boldsymbol{\mu} \times \textbf{B}## and hence the work done is:

$$W=-\int_{\frac{\pi}{2}}^\theta \tau d\theta = - \int_{\frac{\pi}{2}}^\theta \mu B \sin \theta d\theta = \mu B \cos \theta = \boldsymbol{\mu} \cdot \textbf{B}$$

where ##\theta## is the angle between the magnetic moment and the external magnetic field. Thus the potential energy is ##-\boldsymbol{\mu} \cdot \textbf{B}## as required.

But it still isn't clear, what causes this potential energy to exist? Magnetic forces do no work after all.

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