Why is a current ring approx a magnetic dipole from very far off?

[zenterix]

Homework Statement

What exactly does it mean to say that "the magnetic field at a point $r>>R$ due to a current ring of radius $R$ may be approximated by a small magnetic dipole moment placed at the origin"?

Relevant Equations

$\vec{B}=\frac{\mu_0}{4\pi} \frac{Id\vec{s}\times\hat{r}}{r^2}$

The following is from Appendix I of these notes from MIT OCW.

Consider a circular loop of radius $R$ lying in the $xy$-plane and carrying a steady current $I$.

If we compute the magnetic field at a point $P$ off the axis of symmetry in the $yz$-plane we arrive at integrals that can only be computed numerically

Note that the $x$-component of $\vec{B}$ at $P$ is $0$ by symmetry.

Consider a scenario in which $R<<(y^2+z^2)^{1/2}=r$. That is, the radius of the ring of current is much smaller than the distance to the field point $P$.

Using linear approximation for the denominator in the expressions for $B_y$ and $B_z$ above, we reach

and

The notes I am reading say

The quantity $I(\pi R^2)$ may be identified as the magnitude of the magnetic dipole moment $\mu=IA$ where $A=\pi R^2$ is the area of the loop.

Using spherical coordinates, with $y=r\sin{\theta}$ and $z=r\cos{\theta}$ we can rewrite the expressions above as

Finally, the notes say

Thus, we see that the magnetic field at a point $r>>R$ due to a current ring of radius $R$ may be approximated by a small magnetic dipole moment placed at the origin

I really do not understand the concept of a magnetic dipole very well.

Here is what I think I grasp so far

1) A magnetic dipole is a current loop.

2) The magnetic dipole moment is a vector normal to the surface enclosed by the loop (using right-hand rule relative to the current) and has magnitude $IA$.

$$\mu=IA\hat{n}$$

3) The field of a magnetic dipole resembles that of a bar magnet. The bar magnet is also referred to as a magnetic dipole.

Ok, we have these three things for our current ring. 3) is a descriptive reason why we would say that the current ring behaves like a magnetic dipole.

But how do we see this in the equations shown above?

The expressions for $B_y$ and $B_y$ contain a $\mu$ factor in them (but also many other factors). How does this show that we can approximate the ring as a magnetic dipole moment?

 

[kuruman]

A magnetic dipole is a current loop.

It is more correct to say that a current loop has a magnetic moment.
Electrons, protons and nuclei also have magnetic moments, yet there are no currents at play.

If you are at distances close to a ring of current, the field looks like the figure below left. If you move to distances much larger than the diameter of the ring, you will not be able to see any details and the field would like a point dipole below right. The arrow shows the direction of the magnetic dipole vector.

A square current loop would look different from the circular loop at close distances but the same (point dipole) at large distances.

 

[TSny]

Using spherical coordinates, with $y=r\sin{\theta}$ and $z=r\cos{\theta}$ we can rewrite the expressions above as

View attachment 341486



How does this show that we can approximate the ring as a magnetic dipole moment?

I'm not sure I completely understand your question. But, note that the results for $B_y$ and $B_z$ above (far from the current ring) have the same mathematical form as the Electric field of an electric dipole derived in chapter 2 of the MIT notes. See equations (2.7.6) and (2.7.7). The only difference is the electric dipole moment vector is along the y-axis while the magnetic dipole moment vector is along the z-axis. So, this is why we say that far from the current ring, the magnetic field is a dipole field.

 

[zenterix]

@TSny

I think I get it now.

We first define electric dipole. We calculate the field due to this dipole.

Next, we consider a magnetic dipole in the form of a current ring. We calculate the field due to this current ring (at large distances from the magnetic dipole) and notice that the equations are essentially the same as those for an electric dipole at large distances.

Relative to the dipole moment vector, both fields look the same.

The only differences in the equations are that

1) Instead of a magnitude of an electric dipole moment ($|\vec{p}|=2qa$, where $a$ is halfway between the two charges) we have a magnitude of a magnetic dipole moment ($|\vec{\mu}|=IA$, where $A$ is the area of the loop).

2) The dipole moment vector goes from positive charge to negative charge in the electric dipole case, and is perpendicular to the current ring in the magnetic dipole case.

Now at this point I just want to talk semantics.

Consider again the sentence

Thus, we see that the magnetic field at a point $r>>R$ due to a current ring of radius $R$ may be approximated by a small magnetic dipole moment placed at the origin.

The current ring has a magnetic dipole moment at the origin (this is just a simple vector defined very specifically). It is not very clear what it means to approximate the magnetic field of the current ring with a magnetic dipole moment at the origin, when this magnetic dipole moment already basically defines the current ring in the first place.

Is the sentence accurate?

Or is "magnetic dipole moment" in that sentence referring to an actual bar magnet?

 

[zenterix]

It is more correct to say that a current loop has a magnetic moment.
Electrons, protons and nuclei also have magnetic moments, yet there are no currents at play.

If you are at distances close to a ring of current, the field looks like the figure below left. If you move to distances much larger than the diameter of the ring, you will not be able to see any details and the field would like a point dipole below right. The arrow shows the direction of the magnetic dipole vector.

A square current loop would look different from the circular loop at close distances but the same (point dipole) at large distances.

View attachment 341491View attachment 341492

When you say "would look like a point dipole" do you mean "would look like an electric point dipole"?

In other words, when we refer to dipole, it seems that an electric dipole is the reference against which the magnetic field of a magnetic dipole is compared.

Therefore, we say that the magnetic field of the current ring when we zoom out looks like the field of an electric point dipole (if the electric dipole moment were the same vector as the magnetic dipole vector, or at least had the same direction).

 

[TSny]

We first define electric dipole. We calculate the field due to this dipole.

Next, we consider a magnetic dipole in the form of a current ring. We calculate the field due to this current ring (at large distances from the magnetic dipole) and notice that the equations are essentially the same as those for an electric dipole at large distances.

Relative to the dipole moment vector, both fields look the same.

Yes.

The only differences in the equations are that

1) Instead of a magnitude of an electric dipole moment ($|\vec{p}|=2qa$, where $a$ is halfway between the two charges) we have a magnitude of a magnetic dipole moment ($|\vec{\mu}|=IA$, where $A$ is the area of the loop).

2) The dipole moment vector goes from positive charge to negative charge in the electric dipole case, and is perpendicular to the current ring in the magnetic dipole case.

And, of course, $\mu_0$ takes the place of $1/\epsilon_0$.

Now at this point I just want to talk semantics.

Consider again the sentence...

Is the sentence accurate?

Or is "magnetic dipole moment" in that sentence referring to an actual bar magnet?

I'm not sure exactly what the author had in mind. But, I believe you can interpret "magnetic dipole moment" in that sentence to refer to a small bar magnet. That is, a system of equal and opposite "magnetic poles" ("north" and "south") separated by a small distance. The dipole moment of this system would be $md$ where $m$ is the magnitude of one of the magnetic poles and $d$ is the distance between the poles.

 

[zenterix]

magnitude of one of the magnetic poles

What is the magnitude of a magnetic pole?

I don't think I've seen this yet.

 

[TSny]

What is the magnitude of a magnetic pole?

I don't think I've seen this yet.

A magnetic pole ("monopole") would be a particle that carries a "magnetic charge" (either "north" or "south"). It would produce a radial magnetic field with a magnitude that falls off as the square of the distance from the pole. Magnetic monopoles have never been found in nature, despite extensive searching. But you can sometimes use them conceptually to model magnetic field sources. For example, you can approximate a small, thin bar magnet as a magnetic dipole consisting of a north and south pole separated by a small distance. The external field of the magnet will be very similar to that of the fictitious dipole.