What is the equivalent of Electric P.E. in Magnetism?

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1. Apr 9, 2015

userunknown

There is an analogy between electric and gravitational potential energy.
$U_g = \frac {GmM}{r}$
$U_e = \frac {kqQ}{r}$

What is the analogous formula in magnetostatics?

Thanks...

2. Apr 9, 2015

gleem

No.Not exactly. Since magnetic monopoles do not exist as such then a similar equation would be hard to get. But we can define a scalar potential which depends on the current loop producing the field such that B=-Vm (Like E = -V) where Vm=(μ0/4π)Ω⋅I and I is the current of the loop producing the field and Ω is the solid angle that the loop subtends at the field point where we wish to determine B. You should find a derivation in any intermediate EM text.

3. Apr 10, 2015

Staff: Mentor

If magnetic monopoles did exist, then there would be a symmetry between electric and magnetic potentials.

If I understand it correctly, in theory magnetic monopoles could exist, or perhaps should exist, but all observations so far show no evidence that they do exist in this universe.

4. Apr 10, 2015

userunknown

This is confusing. You said: magnetic scalar potential is analogous with electric potential.
But some sources say: magnetic vector potential is analogous with electrical potential.

Thanks for the answer but which one is true?

5. Apr 10, 2015

Staff: Mentor

I mean it in the sense of Maxwells Equations:

Where ρ is electric charge density in the first equation and ρ is magnetic charge density in the second equation. The equations look symmetrical in that case, but it just happens that magnetic charge density is zero.

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6. Apr 11, 2015

gleem

I used analogous in the manner that the magnetic scalar potential is related to the magnetic field through the gradient of a potential as is the electric field. The Vector potential obviously is a vector and is related to the magnetic field via a different vector operator i.e. the Curl. The difference between the two magnetic potentials is that the scalar potential is limited to points outside a conductor while the vector potential is not (like the electric potential). I don't think there is a particular question of truth or falsity about this issue. As I said originally "No. not exactly" because there is not an identical function for a magnetic field compared to an electric field since there is no magnetic charge.

7. Apr 12, 2015

userunknown

Thank you.

I have just one more question. Can we derive magnetic potential energy formula from magnetic scalar potential, like we can do in electrostatics?

I'm searching this topic, I have seen this formula, as magnetic potential formula:
$$U = -\vec m\cdot \vec B$$
Can we derive this potential energy formula from scalar potential?

Thanks again for help...

8. Apr 12, 2015

gleem

U = - M⋅B is specific for a magnetic dipole moment in a magnetic field just like the potential energy for an electric dipole moment U= - p⋅E in an electric field.

The actual energy stored in a magnetic field is

Wm = (1/2⋅μ0) ∫v B2

while that for an electric field is

We = (ε0/2) ∫v E2

9. Apr 12, 2015

userunknown

But if we take Vm=(μ0/4π)Ω⋅I as a potential, based on the analogy between B=-Vm and E = -V, there must be a magnetic potential energy formula based on magnetic scalar potential(Vm=(μ0/4π)Ω⋅I) . Right?

10. Apr 12, 2015

gleem

Actually not. At lest not to my knowledge. The derivation of Ue assumes an electric charge, so since there is no magnetic charge the derivation for magnetic PE cannot include any self energy of a "charge " distribution using Vm. However you can calculate a PE of a magnetic field using the vector poteriial since this can include the current distribution like the Ve does for the charge distribution in the electric PE.

11. Apr 17, 2015

stedwards

As an aside, there actually is. But it takes a little cheating, and then the non-physical magnetic charge has to be gauged-away. This is accomplished with a complex vector potential.