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I'm not quite understanding the concept of a magnetic dipole. From what I've seen, we can treat a given conducting loop as a magnetic dipole ONLY in the approximation where we consider its potential (and consequently the resulting magnetic field) in the limit where the distance from the coil is much greater than the size of the coil itself. In this case, the dipole term dominates the multipole expansion and we can disregard the quadrupole and higher order terms.

The idea of computing the torque on a dipole seems to contradict this, as torque acts locally where the approximation described above is no longer valid. It therefore seems to me that we can no longer treat the coil as a dipole, which is the source of my confusion.

Perhaps I'm misunderstanding the whole idea of a dipole, any clarification would be greatly appreciated.

Elio Fabri
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David
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  • Please, be more specific with equations what you do not understand. And did you mean to write "contradict" and not "contract"? – hyportnex Apr 01 '23 at 13:49
  • Yes I did mean contradict thank you. I haven't mentioned any equations as it's more the concept of applying an approximation for something being far away on a local phenomenon that I'm failing to understand – David Apr 01 '23 at 13:53
  • If I were to include them, the relevant equations would be torque = m x B, where m is the magnetic dipole moment. – David Apr 01 '23 at 13:54
  • hint: draw a square current loop in the $(x,y)$ plane and consider the $B_y$ component of the homogeneous magnetic field. What forces are acting on the loop? – hyportnex Apr 01 '23 at 14:15
  • Got it! It was just an issue with terminology, I confused the meaning of dipole referring to any closed current loop with a 'pure' dipole, which returns solely a dipole term in the multipole expansion. – David Apr 01 '23 at 16:40

2 Answers2

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As the OP has pointed out, when we compute the magnetic field generated by a current loop or other localized current configuration, the approximation in which we treat the loop as a pure dipole, and drop all higher multipole moments, will be valid only at a distance that is much greater in magnitude than the length scale of the localized current configuration.

When we are treating the current loop or other localized current configuration as the object that the magnetic field acts on, the condition for approximating it as a pure dipole is different. To discover what that condition is, it is more convenient to work with the potential energy $U$, which can be expressed in terms of the vector potential $A$:

$$ U = - \int J \cdot A \, \mathrm{d}^3 r = - \int J_i(A_i(0) + r_j \partial_j A_i(0) + \frac{1}{2} r_j r_k \partial_j \partial_k A_i(0) + \ldots ) \, \mathrm{d}^3 r $$

where we are expanding $A$ as a Taylor series around the origin, which is assumed to lie within the localized current configuration.

It can be shown that the first term on the right-hand side, $\int J_i A_i(0)$, vanishes, and that the second term is equivalent to $-\mu \cdot B(0)$, with $\mu$ the magnetic dipole moment of the localized current configuration. This is done here.

Notice that the subsequent terms in the Taylor series involve the product of $J$ with $\ell$ factors of $r$ (i.e. something similar to a higher multipole moment, where $\ell = 2$ corresponds to the quadrupole moment and so on) and the $\ell$-fold derivatives of $A$ at the origin, i.e. something similar to the $(\ell-1)$-fold derivatives of $B$ at the origin.

This leads to the following observation: when calculating the potential energy of a localized current configuration in an external magnetic field, we can neglect moments higher than the dipole moment provided that each time we introduce another factor of $r\partial$, the resulting term is much smaller than the previous one; another way of saying this is that the magnitude of each component of $B$ doesn't vary appreciably in the vicinity of the localized current configuration.

In the special case of a uniform external magnetic field, the dipole approximation for the potential energy is exact, and likewise for the torque (which is just the derivative of the potential energy with respect to an angle, with the sign flipped).

Brian Bi
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  • Thanks for the response! So to clarify, in the case where we have a uniform external B field applying the torque, the potential of any given current loop will behave like a dipole in the Taylor expansion, WITHOUT making any approximations. – David Apr 01 '23 at 21:32
  • @David Yes, in that case, all the terms due to the higher multipole moments will vanish, and there are no approximations involved. – Brian Bi Apr 01 '23 at 21:40
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In think you're just thinking about the exterior multipole expansion for large distance $r$ compared with the loop size $r'$, which has terms looking like:

$$\big(\frac {r'}r\big)^l\sum_{m-l}^{m+l}Y_{lm}(\Omega)Y^*_{lm}(\Omega')$$

When $r\lt r'$, you are inside the loop and you need to do an interior multipole expansion, with terms looking like:

$$\big(\frac r{r'}\big)^l\sum_{m-l}^{m+l}Y_{lm}(\Omega)Y^*_{lm}(\Omega')$$

so the higher moment are suppressed.

JEB
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