Is the Biot-Savart Law reversible?

[Subhra]

Please find the attachment and comment about equation (2).

 

[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.

 

[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"?

 

[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 anyone 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.

 

[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?

 

[UltrafastPED]

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

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.

 

[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?

 

[UltrafastPED]

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?

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.

 

[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)?

 

[Dale]

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.

 

[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.

 

[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?

 

[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)?

 

[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?

 

[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.

 

[carrz]

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.

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.

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

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

 

[carrz]

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.

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).

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.

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.

 

[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?

 

[carrz]

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?

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.

 

[Subhra]

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

 

[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.

 

[Jano L.]

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)?

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.

 

[carrz]

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

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.

 

[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?

 

[carrz]

Please find the attachment and comment about equation (2).

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.

 

[Matterwave]

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?

This question has been answered many times already. A static charge in a static magnetic field will not move.

 

[Jano L.]

If the motion be uniform, the magnetic field will be static, else dynamic.

Only if the uniform motion refers to motion of constant electric current in a circuit. If you mean uniform motion of one charged particle, its magnetic field will not be static. It will move along with the particle.

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?

No, No. If the particle is put at rest into the applied magnetic field of other sources, it will not move, because the magnetic force vanishes.

 

[Subhra]

Whether you say "it will be in motion" or "magnetic field of a charge is proportional to its velocity" doesn't make any difference. The first phrase indicates the state of motion and the second phrase indicates the velocity dependence.

So, Biot-Savart law is about creating its own B field due to its motion. This means MOVING CHARGE ->B. I just want to check:

B->MOVING CHARGE

 

[Subhra]

This question has been answered many times already. A static charge in a static magnetic field will not move.

Then tell me whether the magnetic field in Biot-Savart law is "static".

 

[Subhra]

Only if the uniform motion refers to motion of constant electric current in a circuit. If you mean uniform motion of one charged particle, its magnetic field will not be static. It will move along with the particle.

I am sorry, I used the term "uniform" loosely. You are right in this regard.

No, No. If the particle is put at rest into the applied magnetic field of other sources, it will not move, because the magnetic force vanishes.

Can you please clarify, how does the magnetic force vanish?