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?

 

[carrz]

According to Biot-Savart Law if there be a charge in motion, it will create a magnetic field.

Yes. It creates its own magnetic field, it does not experience it itself.

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

Yes.

Now, if a charge particle be placed in a magnetic field, will it move?

Second charge will experience force only if it has velocity greater than zero.

To put it in another way, is the velocity of the charged particle dependent on the applied magnetic field?

You need to be asking about force/acceleration, not velocity really. Anyway, yes, and it also dependent on its own "initial" velocity.

 

[Matterwave]

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

It is static. But this question doesn't have much to do with your previous question.

 

[Subhra]

Second charge will experience force only if it has velocity greater than zero.

So a stationary charged particle will not experience any force near a current element.

 

[Subhra]

It is static. But this question doesn't have much to do with your previous question.

So a charge in motion creates a static magnetic field. Now in this static field place a static charge and forget about the field. Look only a charge (creator of the magnetic field) in motion and a charge at rest and tell me whether the charge at rest will experience a force or not.

 

[carrz]

So a stationary charged particle will not experience any force near a current element.

B1= q1*v1/r^2, B2= q2*v2/r^2
F1= q1*v1 x B2, F2= q2*v2 x B1

Yes. That's what Lorentz force equation says, if v1 = 0 then F1 = 0. No force means no acceleration which means no change in velocity so it stays zero.

You can not invert Biot-Savart law like in the document you posted, because velocity is not an effect, it's a cause. Velocity of a charge is not influenced or caused by its own B field, which is what the second equation in your document suggests, so it's not true.

 

[Matterwave]

So a charge in motion creates a static magnetic field. Now in this static field place a static charge and forget about the field. Look only a charge (creator of the magnetic field) in motion and a charge at rest and tell me whether the charge at rest will experience a force or not.

The Biot-Savart law is valid for a current in a wire. Not for a single moving charge. A single moving charge will also have an electric field, which will accelerate other electric charges.

 

[carrz]

So a charge in motion creates a static magnetic field. Now in this static field place a static charge and forget about the field. Look only a charge (creator of the magnetic field) in motion and a charge at rest and tell me whether the charge at rest will experience a force or not.

It's clear from Lorenz force equation: F1= q1*v1 X B2

Any number multiplied by zero (v1 = 0) will be zero. Can you see?

 

[Subhra]

B1= q1*v1/r^2, B2= q2*v2/r^2
F1= q1*v1 x B2, F2= q2*v2 x B1

Yes. That's what Lorentz force equation says, if v1 = 0 then F1 = 0. No force means no acceleration which means no change in velocity so it stays zero.

You can not invert Biot-Savart law like in the document you posted, because velocity is not an effect, it's a cause. Velocity of a charge is not influenced or caused by its own B field, which is what the second equation in your document suggests, so it's not true.

What about the force of q2 (moving with v2) on q1(even if v1=0)?

 

[Subhra]

It's clear from Lorenz force equation: F1= q1*v1 X B2

Any number multiplied by zero (v1 = 0) will be zero. Can you see?

If v1=0, F1=0 from the above equation. There is no problem in it.

But if B2!=0, v2!=0. Then will the charge q2 exert a force on q1 or not?

 

[carrz]

What about the force of q2 (moving with v2) on q1(even if v1=0)?

F2 = q2*v2 x B1

But first you need to calculate B1 = q1*v1/r^2

So, if v1 = 0 then B1 = 0 and so F2 = 0.

Also, if v2 = 0 then F2 = 0 regardless of B1.

 

[carrz]

If v1=0, F1=0 from the above equation. There is no problem in it.

But if B2!=0, v2!=0. Then will the charge q2 exert a force on q1 or not?

F1 = q1*v1 x B2


B2 > 0 only if V2 > 0.

q1 will experience force relative to q2 only if both v1 > 0 and B2 > 0.

Just plug the numbers into equation and all this becomes obvious.

 

[Subhra]

F2 = q2*v2 x B1

But first you need to calculate B1 = q1*v1/r^2

So, if v1 = 0 then B1 = 0 and so F2 = 0.

Also, if v2 = 0 then F2 = 0 regardless of B1.

F1 = q1*v1 x B2B2 > 0 only if V2 > 0.

q1 will experience force relative to q2 only if both v1 > 0 and B2 > 0.

Just plug the numbers into equation and all this becomes obvious.

I think, you are focused on the magnetic force only. But a moving charge also has an electric field. This means if you have a magnetic field there should be an electric field associated with the particle causing the magnetic field.

Under this electric field, the rest charge should experience a force resulting a motion of the charge.

 

[carrz]

I think, you are focused on the magnetic force only. But a moving charge also has an electric field. This means if you have a magnetic field there should be an electric field associated with the particle causing the magnetic field.

Under this electric field, the rest charge should experience a force resulting a motion of the charge.

Coulomb force can be calculate along with Lorentz force (depending on equation), but you can also calculate electric force vectors separately with Coulomb's law equation, then just add magnetic and electric force vectors together. I think it's better to separate them to see a problem more clearly.

 

[Subhra]

Coulomb force can be calculate along with Lorentz force (depending on equation), but you can also calculate electric force vectors separately with Coulomb's law equation, then just add magnetic and electric force vectors together. I think it's better to separate them to see a problem more clearly.

This is not my answer. However, it seems from your reply that the particle at rest will certainly move under a magnetic field.

You may argue that the motion is due to the electric field of the charged particle creating the magnetic field.

Now create the magnetic field using a permanent magnet and place a charge in the magnetic field of the magnet and answer whether the charge will move.

 

[Subhra]

The Biot-Savart law is valid for a current in a wire. Not for a single moving charge.

It's not true. See Heaviside's electrodynamics.

A single moving charge will also have an electric field, which will accelerate other electric charges.

Whether it is a single moving charge or a current in a wire, it will create a magnetic field which will move a stationary charge.

This means there should be a mathematical way to describe v=v(B).

 

[carrz]

This is not my answer. However, it seems from your reply that the particle at rest will certainly move under a magnetic field.

You may argue that the motion is due to the electric field of the charged particle creating the magnetic field.

Biot-Savart law has nothing to do with electric forces and you haven't mentioned electric fields until just now. Therefore, we were all ignoring electric force because your questions were vague and and it would only complicate explanation.

Now create the magnetic field using a permanent magnet and place a charge in the magnetic field of the magnet and answer whether the charge will move.

Why don't you answer? You know from before there will be no magnetic force. But you said nothing about electric force, so I assume the magnet is electrically neutral, and so there would also be no any electric force on the charge and thus it will not move.

What part could you not answer yourself?

 

[Matterwave]

It's not true. See Heaviside's electrodynamics.



Whether it is a single moving charge or a current in a wire, it will create a magnetic field which will move a stationary charge.

This means there should be a mathematical way to describe v=v(B).

A current carrying wire is still electrically neutral. A single point charge by definition cannot be electrically neutral.

What you are calling the "Biot and Savart law" is one half of a set of approximate equations where v<<c for which this half resembles the actual Biot Savart law. The other half deals with the electric field, which is NOT accounted for by the Biot Savart law.

But you keep repeating your question about putting a stationary charge in a static magnetic field (and only a static magnetic field). This question has been answered MANY times already in this thread. The charge WILL NOT begin to move.

 

[Dale]

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)

But that is already not magnetostatic. As a charge moves it is a time varying current which produces a time varying magnetic field which is described by the Lienard Wiechert and not by Biot Savart.

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.

I am not certain about the invertibility of Biot Savart (I think it is not invertible, but I am not certain). However, if it is not invertible then that means that you can specify some additional constraints such as minimizing total current density or something similar and iteratively calculate a current distribution to produce a given field.

 

[Dale]

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.

If a charge is in motion then it will create a magnetic field, but that is NOT "according to Biot-Savart". A charge in motion creates a time varying magnetic field so Biot Savart does not apply. The Lienard Wiechert potentials describe the magnetic field of a moving charge.

What Biot-Savart says is that if there is a steady (non-time varying) current then it will create a steady (non-time varying) magnetic field.

If the motion be uniform, the magnetic field will be static, else dynamic. The nature of the magnetic field is not important here.

Neither of these statements are true. Uniform motion of a charge creates a dynamic magnetic field. The nature of the field is critical for determining which equations to use, if it is magnetostatic then you can use the simplified Biot Savart else you must use more complicated equations.

Now, if a charge particle be placed in a magnetic field, will it move?

This has already been answered: no, if it starts with 0 velocity then it will experience 0 force and therefore 0 acceleration.

 

[Dale]

So a charge in motion creates a static magnetic field.

No, it does not! A charge in motion creates a time-varying magnetic field given by the Lienard Wiechert potentials.