How accurate is the magnetic field determined by assuming a circular current carrying loop as a magnetic dipole?

Question

I've learnt that a circular loop of area $A$ carrying a current $i$ produces a magnetic moment equal to $iA$ and the field due to the loop can be considered to be due to a magnetic dipole, consisting of two magnetic charges (magnetic monopoles) - positive (north) and negative (south), each of pole strength $m$ and separated by a distance $d$ such that $md=iA$.

By considering a circular loop carrying a current as a magnetic dipole, the magnetic field could be easily calculated at any point using Coulomb's law of magnetism and law of superposition, instead of using the Biot-Savart law. However, this analogy is not completely perfect because magnetic monopoles have not been observed so far. Also, magnetic field lines do not start or end at a particular point, they only form closed loops and this is contrary to the dipole picture.

Further, the magnetic field inside the dipole system is opposite in direction when compared to that of a circular loop. This can be inferred in the central regions in the following diagrams:

^{Left: Magnetic field due to a current carrying circular loop.
Right: Magnetic field due to positive (north) and negative (south) magnetic charges (magnetic monopoles) separated by some distance (magnetic dipole).}

^{Image source: Magnetic dipole - Wikipedia}

It can be seen that the magnetic field lines are similar in both diagrams except in the central region where the directions are opposite. Also, the field at the assumed pole cannot be determined by using Coulomb's law of magnetism as $r=0$ in $\frac{\mu_0}{4\pi}\frac{m}{r^2}$. However, it can be easily calculated using Biot-Savart law.

My questions are as follows:

1. Is the magnetic dipole picture of assuming a circular loop carrying a current is a kind of "approximation"? Or in other words, is the field determined by assuming magnetic fields due to magnetic monopoles different from the original field due to circular loop?
   In short, how accurate is the magnetic field determined by assuming a circular current carrying loop as a magnetic dipole?

2. Does the pole picture causes variation only in the direction or also includes difference in the magnitude of the field?

3. Where does the pole picture give accurate result for both magnitude and direction of the magnetic field around a circular loop carrying a current?

4. What are the specifications for the choice of $m$ and $d$? From $md=iA$, the product $md$ is a constant for a particular $i$ and $A$, however, we're free to choose $m$ and $d$ in such a way it satisfies the condition. So does a small value of $d$ (and large value of $m$) give more accurate results, or is it the other way round?

Answer 1

I feel like a lot of these questions are asking the same thing, so do tell me if I've misinterpreted any.

1. Is the magnetic dipole picture of assuming a circular loop carrying a current is a kind of "approximation"? Or in other words, is the field determined by assuming magnetic fields due to magnetic monopoles different from the original field due to circular loop?

Yes. For example, as you can clearly see in the figure, the two fields don't even point in the same direction in the interior.

In short, how accurate is the magnetic field determined by assuming a circular current carrying loop as a magnetic dipole?

It gets more accurate the further away you get, because both of them approach the ideal dipole field, $$\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4 \pi} \left(\frac{3 \hat{\mathbf{r}} (\mathbf{m} \cdot \hat{\mathbf{r}}) - \mathbf{m}}{r^3} \right).$$ More precisely, at large distances, both of these fields differ from the ideal dipole field, in both magnitude and direction, by a fractional amount proportional to $\ell/r$, where $\ell$ is a characteristic length scale. For the loop picture, $\ell = \sqrt{A}$. For the pole picture, $\ell = d$.

As you go to smaller $r$, the divergences get a lot bigger, e.g. for $r \lesssim \ell$ the fields are completely different from the ideal dipole field, and from each other.

2. Does the pole picture causes variation only in the direction or also includes difference in the magnitude of the field?

Yes. For example, as you can see from the figure, the magnitudes diverge at the poles and at the loop, and there's no corresponding divergence in the other picture.

3. Where does the pole picture give accurate result for both magnitude and direction of the magnetic field around a circular loop carrying a current?

At large $r$.

4. What are the specifications for the choice of $m$ and $d$? From $md=iA$, the product $md$ is a constant for a particular $i$ and $A$, however, we're free to choose $m$ and $d$ in such a way it satisfies the condition. So does a small value of $d$ (and large value of $m$) give more accurate results, or is it the other way round?

What is exactly true is that in the limit $d \to 0$ and $A \to 0$, the two fields approach each other, because they both become the same as the ideal dipole field.

When $A \neq 0$, it's not clear what value of $d$ gets a more "accurate" result, because it depends on how you define "accuracy". For example, if you wanted the field at $r = 0$ to be as accurate as possible, then you should send $d \to \infty$, because that gets zero field at $r = 0$. But then that would get the field at $r \approx \sqrt{A}$ totally wrong. If an IIT JEE problem asks you what the most "accurate" result is, this is a vague and undefined criterion and the correct answer is whatever random thing the test writer had in mind at the moment.

Answer 2

If the goal is to calculate the field of a finite ring, there is no point in using a pair of monopoles to approximate it. The real purpose is to show that, in the limit as the loop shrinks to zero diameter, it approaches the field of a pure dipole with the same magnetic moment. (See this earlier Stack Exchange question.) The Wikipedia article on magnetic dipoles derives expressions for the internal fields in that limit, and it is interesting that you can define $B$, $H$ and $M$ so that $B = \mu_0 (H+M)$. However, we're talking about the internal field of a point dipole, so there is no practical application.

The practical use of fictitious monopoles is for calculating the internal field of a magnetic material, known as the demagnetizing field. The magnetization can be expressed in terms of current loops, but the calculations are far easier when you express it in terms of monopoles. And they lead to exactly the same result. Also, in a ferromagnet or other magnetically ordered substance, the magnetization is almost entirely due to electron spins, so a representation by currents is no more realistic than one using monopoles. For these reasons, experts on magnetism always use the monopole approximation.

In these applications, a magnetic charge density is defined as proportional to $\nabla \cdot \mathbf{M}$ inside the magnet and $\mathbf{M}\cdot \mathbf{n}$ (the surface normal component of the magnetization) at the surface. These arise naturally from applying Maxwell's equations with no additional assumptions; it's just a particular way of solving them. See the Wikipedia articles on the demagnetizing field (linked above) and micromagnetics for more details.