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

[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...

 

[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=-∇V_m (Like E = -∇V) where V_m=(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.

 

[anorlunda]

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.

 

[userunknown]

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=-∇V_m (Like E = -∇V) where V_m=(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.

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?

 

[anorlunda]

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.

 

[gleem]

But some sources say: magnetic vector potential is analogous with electrical potential.

Thanks for the answer but which one is true?

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.

 

[userunknown]

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.

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...

 

[gleem]

I'm searching this topic, I have seen this formula, as magnetic potential formula:

U=−m⃗ ⋅B⃗​

U = -\vec m\cdot \vec B
Can we derive this potential energy formula from scalar potential?

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

W_m = (1/2⋅μ₀) ∫_v B²dτ

while that for an electric field is

W_e = (ε₀/2) ∫_v E²dτ

 

[userunknown]

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

W_m = (1/2⋅μ₀) ∫_v B²dτ

while that for an electric field is

W_e = (ε₀/2) ∫_v E²dτ

But if we take V_m=(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(V_m=(μ₀/4π)Ω⋅I) . Right?

 

[gleem]

Actually not. At lest not to my knowledge. The derivation of U_e 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 V_m. However you can calculate a PE of a magnetic field using the vector poteriial since this can include the current distribution like the V_e does for the charge distribution in the electric PE.

 

[stedwards]

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.

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.