# Why is the magnetic field of a wire circular?

1. Aug 18, 2007

### Mr Virtual

Hi
First of all, since magnetic field in a wire carrying current is due to the movement of electrons, I assume that the magnetic field of a single isolated moving electron, or any other charged particle, can also be deduced by the right hand thumb rule i.e. the field will be much like Saturn's ring around it, where we assume saturn to be a charged particle, and its ring as its field.

Now my question is: How is this field deduced. Why is it circular? Is it the result of superposition, if any?

Secondly, I know that magnetic force exerted on a moving charge consists of a cross product of B and v. But I think that this force is the result of interaction between the charge's own magnetic field and the applied field, B.
My question is: How does this interaction exactly happen? How does it result in this force? Most importantly, what causes this force to be perp to both B and v (I need an explanation other than that it is just the result of cross product)? I raised a similar question elsewhere but didn't get much help.
I'd be grateful if somebody explains.

Thanks
Mr V

2. Aug 20, 2007

### lugita15

The proof that magnetic field lines due to a straight, infinitely long, constant current are circular is based on the Biot-Savart Law.
The Biot Savart Law states that the magnetic field dB produced by an infinitesimal segment of wire ds which carries a current I through it is given by

The proof itself involves quite messy vector calculations, but amounts to this: proving that at any point P, B is orthogonal to a radius vector r, drawn perpendicular to the wire with initial point on the wire and terminal point at P.

3. Aug 20, 2007

### Pythagorean

It's not always circular.

I don't know if there's a particular answer to 'why' it happens. Physics examines 'what' happens. When a charge moves, it creates a magnetic field, and the magnetic field is always arranged perpendicular to the direction of motion of the charged particle and perpendicular to the Electric field, following the right-hand rule as you mentioned.

Given those rules, a line charge moving in a direction along the length of itself will generate a circular magnetic field. There are cases where a straight field can be generated too (like inside a coil).

I suppose you could see it as a superposition of the induced field from all the point charges that make up the distributed line charge (the wire), but point charges in electrodynamics are more complicated than distributed charges.

4. Aug 20, 2007

### mgb_phys

Another way of looking at it is just symmetry, if you have a field from a long straight wire what other shape could it be?
If it was square - where would the information come from to specify where the corners should be? This also explains why the field from a point charge should be spherical.

5. Aug 20, 2007

### lugita15

Well, the magnetic field lines, in principle at least, could be going outward from the wire. The actual proof the magnetic field line are circular must be derived from Ampere's Law or the Biot-Savart Law. Symmetry arguments alone aren't enough.

6. Aug 20, 2007

### lugita15

The observation that the magnetic field created by a moving charge is perpendicular to the direction of motion is a consequence of the Biot-Savart Law, and is what the OP wanted proved.

7. Aug 20, 2007

### cesiumfrog

Symmetry is enough. Since reversing the current in a loop is no different from looking at it from the other side, the magnetic field cannot symmetrically have any outward component.

Then again, plenty of students would have asked "but why does it have to be symmetric?". Can you say there are no cases of spontaneous symmetry breaking in physics?

8. Aug 20, 2007

### pardesi

well symmetry arguements are helpful and indeed too good but are useless(here for other components) without bioy-savart law.
this example may be stupid but anyway-suppose i newly discovered the field due to wire and suppose i say the field due to wire at any point depends on say how i look at it ,or depends on the sorce that produces the current...which kill symmetry arguements then u can't decide.
well what i am supporting is well cooperated operation of biot savart and symmetry and in general physical laws(through observations) and symmetry

9. Aug 21, 2007

### stmartin

And why it is not dipole? I think it is not circular, it is not correct.

10. Aug 21, 2007

### Mr Virtual

Can I now get an answer to: how the force felt by a moving charge in a magnetic field is perpendicular to both B and v. I mean any explanation besides that cross product rule is involved here?

Mr V

Last edited: Aug 21, 2007
11. Aug 21, 2007

### stmartin

Can you tell me why it is not dipole, please?

12. Aug 21, 2007

### Pythagorean

who said that and what is "it"?

13. Aug 21, 2007

### stmartin

Why it is circular, why not dipole? On all of the posts, noone exactly said what is it. Please let's define it correctly. Thank you.

14. Aug 23, 2007

### lugita15

I can't understand what you're saying. Are you asking why the magnetic field lines are circular? The Biot-Savart Law, like I said before. I'm not sure what you mean by "why not dipole". Why aren't the magnetic field lines of an infinitely long, straight current the same as those of a magnetic dipole? There's no a priori reason they should be.

15. Aug 25, 2007

### stmartin

And why there is prior that, the field should be circular?

16. Aug 25, 2007

### genneth

Assume the following -- interactions occur because of fields, and do a simple experiment: set up two long wires with currents, and observe that they attract for parallel currents, and repel for antiparallel ones, and also that the force is purely perpendicular to the wires. Assume that this is a reasonable model for infinitely long wires.

By symmetry, you'll deduce that the field is circular about the wire. And you'll also deduce, by symmetry, that the force on a current is perpendicular to the current and the field. Try thinking up of the arguments yourself -- I'll post the full logical steps in a few days.

17. Aug 25, 2007

### jostpuur

Why magnetic field has the shape it has is a small problem compared to the question what magnetic field itself is. This is not a mere philosophical issue, the question "what magnetic field is" has some answers that come from special relativity. Mr Virtual, do you understand what I say in the post #9 in this thread https://www.physicsforums.com/showthread.php?t=175438 It could be true that I start the post too provocatively. It's best to ignore it, I agree mostly what pervect says later in the thread.

I don't think that's correct. It is in fact the charge that interacts with the fields.

You could get, but the answer is technical, so I'm not starting this with LaTex right away. I can start typing more details of the calculation if you are with me in my post #9 of the thread linked above.

18. Aug 28, 2007

### stmartin

Last edited by a moderator: May 3, 2017
19. Aug 28, 2007

### Staff: Mentor

The first is a magnetic dipole field. The second is the field produced by a long straight wire (current). If by "equal" you mean "same configuration", the answer is "no". The two configurations of currents are different, so the fields are different.

20. Aug 28, 2007

### stmartin

And what is the difference? In the both cases the electrons are moving.