Is the Biot-Savart Law reversible?

1. Jul 11, 2014

Subhra

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2. Jul 11, 2014

UltrafastPED

How would you go about proving/disproving equation (2), given that equation (1) is a true statement of the Biot-Savart law?

Hint: start by verifying that the magnitude is correct/incorrect.

3. Jul 11, 2014

Subhra

I am not sure about equation (2). If you consider the magnitude only, equation (2) seems to be fine, however, it mat not be true always quantitatively. But that's not my concern. What I try to observe is only the physical aspects of eqn. (1) and (2).

If B=B(v), can it be infered that v=v(B)?

To put it in another way,

If eqn(1) implies that "if there be a charge in motion there be a magnetic field", can it also infer that "if there be a magnetic field, there be a motion of a charge"?

4. Jul 11, 2014

UltrafastPED

B exists over a region of space related to the charge q by a radius vector.

When the "inversion" takes place it is not so clear what the meaning of r is; there are many points in the field B, but only one velocity for the charge q at any one of them.

That is, it seems to require a many-to-one inversion. So I am not at all clear on what it is supposed to mean.

Perhaps more context for the problem would clarify this.

5. Jul 11, 2014

Subhra

I didn't even tried to understand the meaning of r. This probably implies that if there be a magnetic field at a point P1 and the charge at P2 such that P2-P1=r, the velocity of the charged particle will be v due to the field at P1.

My question is very simply: If a moving charged particle can create a magnetic field, can a charged particle move in a magnetic field?

6. Jul 11, 2014

UltrafastPED

Sure ... the field created by the particle is independent of the field that it is moving in; that is, it doesn't "feel" the field which it creates.

7. Jul 11, 2014

Subhra

I am sorry, I didn't understand you. Consider a charged particle at rest. Now place a bar magnet close to it. What will be the equation of motion of the particle due to the applied magnetic field?

8. Jul 11, 2014

UltrafastPED

The Lorentz force law: F = qv x B.

There is no force if the charges are stationary, and if the magnetic field is static. But when you move the magnet, their is a relative velocity, so there will be a force.

9. Jul 11, 2014

Subhra

If you consider Lorentz force Law, no force will act. Is it so if you consider Biot-Savart Law or in this case equation (2)?

10. Jul 11, 2014

Staff: Mentor

Biot Savart is not a law of nature, it is an approximation which is only valid for magnetostatic situations. It does not apply when the magnetic field changes over time.

The Lorentz force law and Maxwell's equations are the general (classical) laws. When Biot Savart disagrees with them (and it does sometimes) then Biot Savart is wrong.

11. Jul 12, 2014

Subhra

Interesting Answer. By the way, in this case the situation is magnetostatic. Furthermore, I have assumed only the qualitative argument (If there be any motion of a charge, there will be a magnetic field) to be true and tried to know whether the counter argument i.e. if there be any magnetic field, there will be a motion of a charge is true.

12. Jul 12, 2014

Matterwave

Maybe you can start with a simpler problem. If I know that $\vec{A}=\vec{B}\times\vec{C}$. Can I, in general, obtain B uniquely from A and C (without a priori knowing anything about B)? What does a cross product do?

13. Jul 12, 2014

Subhra

I know this trick of vector product. But this won't resolve the problem. The problem is still on the causality relationship: A->B => If A then B; A<->B=> A->B and B->A. Therefore, my question is still: Will there be a motion of a charge in magnetic field (according to Biot-Savart Law)?

14. Jul 12, 2014

Matterwave

My point is that no such unique B can be found because many different B's will create the same A with the same C.

How did you "invert" the Biot-Savart law?

15. Jul 12, 2014

Subhra

Due to the cross product, there might not be any unique solution for the velocity of the charged particle. Yes, I agree to it. If this is so, there should be at least one non-zero value of the velocity.

I don't find any way to "invert" the Biot-Savart Law MATHEMATICALLY and that's why I didn't defend equation (2). I am just trying to find the causality relationship between the magnetic field and the motion of a charged particle. In this regard, Biot-Savart law tells us that if a charge particle moves, it will create a magnetic field and I wish to know whether the reverse is true or not i.e. can a magnetic field move a charged particle or not.

16. Jul 12, 2014

carrz

That's right. There are always two charges. To solve for more than two, force is calculated between each pair. To calculate force on P1 use B field of P2. Then "reverse", that is calculate force on P2 by using B field of P1.

P1 moves in B field of P2, and P2 moves in B field of P1.

17. Jul 12, 2014

carrz

There is always unique solution. If B field of P2 is zero (zero velocity), then P1 will simply not experience any force/acceleration relative to P2, but P2 will experience force relative to P1 if B filed of P1 is greater than zero (non-zero velocity).

Calculate Lorenz force for P1 by using Biot-Savart law for P2. Then "reverse" and calculate Lorenz force for P2 by using Biot-Savart for P1.

18. Jul 12, 2014

Subhra

The point is misunderstood here. Let's have a magnet at point A. Now put a charge at point B near A. Now come to the question: Will the charge move?

19. Jul 12, 2014

carrz

Lorenz force of that charge is proportional to its velocity, so the charge will experience acceleration only if its velocity is greater than zero. But note the charge will experience the same B field regardless of its own velocity and only relative to distance. Direction of acceleration will depend on cross product between B field of the magnet and velocity vector of the charge.

20. Jul 12, 2014

Subhra

So, according to you a charge at rest will remain at rest even if there be a magnetic field?

21. Jul 12, 2014

vanhees71

I don't understand the question, nor the content of the word document in #1 (one shouldn't look at word documents containing physics/math texts anyway, but that's another story). The Biot-Savart-Law only applies to stationary currents, fulfilling the continuity equation, which in this case reduces to $\vec{\nabla} \cdot \vec{j}=0$.

A moving charge does not provide a stationary current, and thus you have to use the retarded potentials or Jefimenko's equations for the fields (which are also retarded solutions).

For the special case of a charge moving with constant velocity, you can use a Lorentz boost to the restframe of the particle, solve the then electrostatic problem there and boost back to the original frame to get the electromagnetic field of a uniformly moving point charge.

22. Jul 12, 2014

Jano L.

UltrafastPED answered your question in post #8. Due to the Lorentz force law for magnetic field
$$\mathbf F = \mathbf v\times \mathbf B,$$
no matter what the form of the magnetic field $\mathbf B$ is, the magnetic force on stationary charged particle is zero. The particle will begin to move only if there is electric field or another force. Only those can accelerate stationary particle.

23. Jul 12, 2014

carrz

According to Lorenz force.

http://en.wikipedia.org/wiki/Lorentz_force

Because there must be two charges the equation should go like this really:

1. Force on P1: F(p1) = q(p1)[E(p1) + v(p1) X B(p2)] "normal"
2. Force on P2: F(p2) = q(p2)[E(p2) + v(p2) X B(p1)] "reverse"

B field in Lorenz force equation is always of the other charge and proportional to that other charge velocity.

24. Jul 12, 2014

Subhra

The question is very simple. I repeat:

According to Biot-Savart Law if there be a charge in motion, it will create a magnetic field. (Look into Heaviside's electrodynamics to confirm the existence of this law for point charges). If the motion be uniform, the magnetic field will be static, else dynamic. The nature of the magnetic field is not important here.

Now, if a charge particle be placed in a magnetic field, will it move? To put it in another way, is the velocity of the charged particle dependent on the applied magnetic field?

25. Jul 12, 2014

carrz

DOCX: "This will mean that if a charged particle be in magnetic field, it will be in motion."

That is not correct. It should say magnetic field of a charge is proportional to its velocity. Biot-Svart law is not about a charge being in magnetic field, it's about it creating its own B field, it does not experience it itself. Lorenz force equation is about charge experiencing magnetic field of another charge.