2 Types Of Magnetic Potential Energy

[sawer]

There are 2 types of magnetic potential energy equations:
1. ##U = -\vec \mu \cdot \vec B##
2. ##U = \frac{1}{2} \int \mathbf{A} \cdot \mathbf{J} \, \mathrm{d}V##

- I have searched for the second equation, only can find some information in one web site. Do you know what their names are and differences?

- I see that second energy equation is derived from magnetic vector potential. But for the first equation, which potential equation is it derived from? Is it magnetic scalar potential?

 

[Hesch]

Please explain the symbols: U, μ, A, J, V.

 

[marcusl]

The first equation simply measures the mechanical work done by rotating a magetic dipole μ in a uniform magnetic feld.
The second equation comes from the expression for field energy [itex]W=\int{\vec{H}\cdot\vec{B}}dV[/itex]. I`m looking at the derivation from that to your equation given in Sommerfeld`s book (the only one I have handy at home) but you should find it in any upper level undergrad book.

 

[marcusl]

Please explain the symbols: U, μ, A, J, V.

Energy, magnetic moment, vector potential, current density, volume. If you don't know what they mean, you`ll need to study a little E&M first.

 

[stedwards]

The first equation simply measures the mechanical work done by rotating a magetic dipole μ in a uniform magnetic feld.
The second equation comes from the expression for field energy [itex]W=\int{\vec{H}\cdot\vec{B}}dV[/itex]. I`m looking at the derivation from that to your equation given in Sommerfeld`s book (the only one I have handy at home) but you should find it in any upper level undergrad book.

The first part looks good, but notice that the second equation contains the current density. I think [itex]A \cdot J[/itex] could be associated with the kinetic energy, and [itex]H \cdot B[/itex] the potential energy. These terms appear in the electromagnetic lagrangian as the magnetic components of A^J and F^*F . I'm not sure how the various tensor elements divide into potential and kinetic energy.