# Moving Charges and Magnetic Moments

1. Feb 11, 2013

### quantumfoam

Does any moving charge generate a magnetic moment? I thought so because a moving charge generates a magnetic field.

2. Feb 12, 2013

### quantumfoam

I know that charges moving in a circle generate a magnetic moment, but I was wondering if charges travelling in a straight line also generate a magnetic moment.

3. Feb 13, 2013

### quantumfoam

If a moving charge travelling in a straight line generates a magnetic field, wouldn't it also have a magnetic moment?

4. Feb 14, 2013

### deep838

Well, consider the dipole moment in electrostatics... its created by placing two charges... So we can say that there will be a magnetic moment if there are two magnetic poles. Now when a charge is moving in a closed path, it creates a magnetic field similar to what would have been created if two poles had been there instead of the charge. What I mean is that when a charge rotates, it creates two poles, North in the anticlockwise sense and South in the clockwise. So we get a magnetic moment in this case. But when a charge is moving in an open path, it doesn't create these to poles and hence we get no magnetic moment.

Thats what I think. Let's wait for others' opinions as well.

What do you think?

5. Feb 14, 2013

### quantumfoam

I think since a magnetic field should have a magnetic moment, which you can use to calculate the magnetic field, a moving charge no matter what kind of motion, generating a magnetic field, should have a magnetic moment.

6. Feb 15, 2013

### deep838

Well, you don't always need the magnetic moment to calculate the mgnetic field. Consider a straight wire! We can find the field at a distance by simple integration from Biot-Savart's Law! Moreover we have Ampere's Circuital Law for a more general case. These don't require the knowledge of magnetic moments.

7. Feb 16, 2013

### quantumfoam

But if I wanted to, would I be able to describe the magnetic field of a moving charge in terms of its magnetic moment?

8. Feb 17, 2013

### deep838

I don't think anyone can give a negative answer to that! If you are capable enough, then certainly you can! But I can't help you with that. Talk to someone whom you know... like your teachers and Head of Departments... and if you get an answer from them, positive or negative, post it here so that I can know as well.

9. Feb 17, 2013

### TopiRinkinen

Taking a charge on circular path, the moment at the centre of rotation is calculated by

$\vec m = I*\vec A$
(A=circular area)

And in the case of single charge the current is:

$I= \frac{dQ}{dt}$
where
$t=\frac{2\Pi r}{v}$
and the moment becomes:
$\vec m=\frac{\vec r \times \vec v Q}{2}$

Now if we start incrasing the size of cicular path towards ∞, the moment increases linearly. But the "reference point" of the moment approaches infinite distance from the charge.

Note that the moment itself is not a field; all calculations using moments are done assuming a moment on definite point.

Another formula for the moment is:
$\vec m=\frac{I}{2}\int \vec {r} \times \vec {dr}$

Which can be used to calculate magnetic moment at origin when the conductor is at (a,y) (vertical line having distance a from origin):

$\vec m = \frac{I}{2}\int ^{\infty}_{-\infty}(a\vec i+y\vec j)\times(\vec j dy)$
$\vec m = \frac{I}{2}\int ^{\infty}_{-\infty}a\vec i\times\vec j dy$
$\vec m = \frac{aI}{2}\vec k \infty$

Thus the magnetic moment for a straight line, calculated at distance a from the line, is infinite, if a>0.

An analogy in mechanics would be calculating the moment of inertia of a disc at point which is NOT the center of mass. Which seems ridiculous.

I would say that the magnetic moment is a property of a system, and valid only at the "centre" of the system (how to define the centre is another problem). As it is not a field, it cannot be used as such for calculating field quantities.

BR, -Topi

10. Feb 17, 2013

### deep838

Why do you say that finding moment of inertia about an arbitrary point is ridiculous??? It's perfectly valid and there are well-defined moment of inertia's about any axis at any point in a system!!

And about the magnetic moment, why do say that it is valid only at the "center"? Is it not distributed over the entire surface? I mean any arbitrarily coiled loop carrying charges have well defined magnetic moments! You said it yourself... the product of the current and the area...!!

11. Feb 17, 2013

### quantumfoam

The magnetic moment points from the south pole to the north pole of a magnet, which I believe to imply magnetic dipoles. If a moving point charge generates a magnetic field according to the Biot-Savart law, (which shows the two magnetic poles of the moving point charge) shouldn't the moving charge also generate a magnetic moment regardless of what type of motion it is executing? Please correct me if I am wrong.

12. Feb 17, 2013

### deep838

But the field due to a "not coiled wire", like a straight wire, isn't separated as north or south. The field rewinds back on itself after 360°. I mean, the field due to a straight wire is curved in the shape of a circle. The direction of the field at a point is a tangent to this circle, at that point. Hence the field itself changes direction from one point to another. What will appear as north pole at one point, will appear to be south pole at the point which is 180° from the previous point. So how can you say that there is a well-defined north or south pole?

13. Feb 18, 2013

### quantumfoam

Yes, you are right about the field appearing to be circular, but isn't the field zero in the direction of velocity of the charge? This would give our moving charge a kind of "donut" shape of some sort. We can't define which one is south or north, but we can sure see the poles

14. Feb 18, 2013

### deep838

I don't understand what you mean by that. Could you elaborate? If you can't "define" the north or south, then how can you "see" the poles?

15. Feb 18, 2013

### quantumfoam

What I meant by "define" is that it depends on the observer on what he chooses to be the south and north pole. The way you can see that the poles are there is by seeing the two opposite points that have no field at them. I am very sure you are familiar with the Biot-Savart equation. According to the equation, the magnetic field of the moving point charge in both the direction of the velocity and in the direction opposite of the velocity is zero, hence the "donut" shape of the magnetic field. Since it has two endpoints, (which, by the way, i'm assuming implies magnetic dipoles) then the magnetic moment of the moving charge can be defined since it is a constant vector pointing from the south pole to the north pole of the magnetic field. Again, this horrible way of seeing it like this may be wrong, so please correct me. If it makes sense, please let me know

16. Feb 18, 2013

### deep838

I still don't understand this... the field is in the shape of a cylinder with the wire as it's axis.

Even if I take it to be a donut, where do you find endpoints in a 'donut'?

That's what I pointed out earlier... If the direction of the field keep on changing with position, how can there a fixed north pole for the magnetic moment to point at!!!

I am sorry if my answers are not good, but honestly, I can't visualize what you are trying to say! I just can't see the two poles at two distinct places throughout the spread of the field and hence i can't see any sign of the magnetic moment.

I am really sorry that after almost a week, I couldn't provide you with a satisfactory answer! :(

17. Feb 18, 2013

### quantumfoam

Nooooo! I don't think we are on the same page here is all. (: I don't think imagining our charge to be a wire is going to help because a wire and a point charge have different magnetic field shapes. What I meant by the donut shape of our point charge is that it is not completely in the shape of a donut, but it has some characteristics. In a real donut, there is a hole, which I totally understand why you got confused (that was my mistake by not being clear enough), but in the magnetic field of our point charge there is no hole. It has the same curving features of a donut but it doesn't have a whole. It has the same magnetic field shape as a solenoid. Our moving point charge is moving at a constant velocity and through a vacuum (classical vacuum). If it is not accelerating or near any force, how will the direction between north and south change?

Last edited: Feb 18, 2013
18. Feb 18, 2013

### quantumfoam

A point charge moving at a constant velocity will have the same magnetic field shape as a bar magnet.

19. Feb 18, 2013

### deep838

Oh!!! Now I get what you want to say.... Well then, if that is the case, then surely the magnetic field has a constant direction and hence a fixed north and south. Now we can safely ask whether there's a moment or not. But there's still one problem... the distance between the two poles! How do you get that?
Probably not possible, but there's a detour.
This donut shaped motion of the charge will be equivalent to current flowing through a coiled wire. And we all know the magnetic moment in such a situation.

20. Feb 18, 2013

### quantumfoam

Right! That is what I was thinking! Thank you very much deep838!! (:

21. Feb 18, 2013

### Staff: Mentor

No. They are very different.

Do a Google search for "magnetic field of a moving point charge" and "magnetic field of a dipole".

22. Feb 19, 2013

### deep838

but then again, coming back to your original question, i don't think that any type of motion will have a magnetic field!
the charge in circular motion is a very special case!

23. Feb 19, 2013

### chill_factor

yes moving charges produce a magnetic field.

you can explicitly see this when you have a static charged particle in one frame moving relative to another frame. in relativity, neither electric fields nor magnetic fields are invariant in different frames, but the electromagnetic tensor is. You take the Lorentz transformation of the electromagnetic tensor that describes a static charged particle in the rest frame in order to transform it into the moving frame and write out the terms explicitly.

You will find that the B'$_{i}$ terms in the moving frame will have nonzero components related to the E field in the rest frame.

24. Feb 20, 2013

### deep838

oops! sorry i wrongly wrote "magnetic field"!!!!

i meant "magnetic moment".

really sorry for that.

So, does any type of motion have a magnetic moment?

25. Feb 23, 2013

### quantumfoam

Actually jtbell, I did the search and they look exactly the same. The only difference is that a point charge has an infinitetismally small distance between it's two magnetic poles.