The Hall Effect vs. The Lorentz Force

[uby]

Hello all,

I am having a slight conceptual problem in rationalizing what happens when a current-carrying material is exposed to an orthogonal magnetic field.

The Lorentz force experienced by the moving charged particle is generally given by (dropping directional notations for clarity):
F = q(E + v x B)

In the absence of an applied electric field, this simplifies to:
F = q(v x B)

In the classical case of a current-carrying wire, this force results in a deflection of the wire. In other words, there is a NET force acting on the charges moving through the wire.

There is also an effect called the Hall Effect, in which an electric field is induced due to the magnetic force:
E = -(v x B)
This field produces a voltage which can be measured.

However, if the force caused by this field is truly in opposition to the Lorentz force, shouldn't the net force on moving charge carriers be ZERO?
i.e.) F = q(E + v x B) = q(-(v x B) + v x B) = 0

Why would the wire deflect if the net force on the carriers is zero? Is it because the induced electric field creates a force on the non-mobile charges in the wire (i.e. - metal cations)?

Thanks

 

[mburt]

F = q(E + v x B) = q(-(v x B) + v x B) = 0

This statement doesn't mathematically make sense, unless the charge is zero in each case. I'm not an expert on these laws, but given factors F = ab, if F = 0 (net zero force) then ONE of those factors must be zero (either charge [q] or [E + v x B]).

I'm not sure if that helps at all, but I was curious about this thread! This is just a general statement I am making.

 

[uby]

This statement doesn't mathematically make sense, unless the charge is zero in each case. I'm not an expert on these laws, but given factors F = ab, if F = 0 (net zero force) then ONE of those factors must be zero (either charge [q] or [E + v x B]).

I'm not sure if that helps at all, but I was curious about this thread! This is just a general statement I am making.

Perhaps omitting vector notation was not a good idea. F, E, v and B are vectors (q is a scalar) with x being the vector cross-product such that (v x B) is also a vector. An electric field (the Hall field) is induced by the magnetic force that is equal but opposite q(v x B) = -q(E) to satisfy equilibrium. In other words, the net force on the current carrying charges in the direction orthogonal to the direction of current flow and B field is zero (else, they'd be flying off the conductor!).

But, if the force on these mobile charges in the direction of deflection is zero, then there must be another force causing the Lorentz deflection. My best explanation is that the Lorentz force magnitude is the sum of the forces on the non-mobile charges (for a metallic wire, the metal cations) caused by the induced electric field. Since they are non-mobile, no induced fields can counter-act the electric field and thus they translate until offset by 'frictional' lattice forces.

 

[Studiot]

My best explanation is that the Lorentz force magnitude is the sum of the forces on the non-mobile charges (for a metallic wire, the metal cations) caused by the induced electric field. Since they are non-mobile, no induced fields can counter-act the electric field and thus they translate until offset by 'frictional' lattice forces.

Firstly please note that for non moving charges v is zero in the equation so the force is zero.

A search will reveal that this question has been discussed many times on PF.

I posted the full expansion of the vector equations in post#42 of this thread for instance.

https://www.physicsforums.com/showthread.php?t=383573&highlight=lorenz+force&page=2

 

[vanhees71]

There's a lot of confusion in this thread. It's always good to go back to the fundamental equations, which in this case are the Maxwell equations and the Lorentz force.

The Hall effect in the most simple form refers to the following setup: You have a conductor along which you apply a voltage resulting in a current. Then you apply a magnetic field perpendicular to the plane of the conductor. Then on each electron there acts a Lorentz force $$\vec{F}=-e \vec{v}\times \vec{B}$$. Thus the electrons drift in the corresponding direction. In this way, a negative surface charge is piled up at the end of the conductor in this direction and a corresponding positive surface charge on the opposite end. This in turn causes and electric field. After some time, you are back in a stationary state, where the force of the electric field, created through the pileup of surface charges, and the Lorentz force from the magnetic field of the electron precisely cancel each other. Through measuring the voltage drop according to this induced electric field you can (a) find out that it are indeed negative charge carriers that make the current (since the electric field would be opposite for positve charges) and (b) the density of conduction electrons.

 

[uby]

There's a lot of confusion in this thread. It's always good to go back to the fundamental equations, which in this case are the Maxwell equations and the Lorentz force.

The Hall effect in the most simple form refers to the following setup: You have a conductor along which you apply a voltage resulting in a current. Then you apply a magnetic field perpendicular to the plane of the conductor. Then on each electron there acts a Lorentz force $$\vec{F}=-e \vec{v}\times \vec{B}$$. Thus the electrons drift in the corresponding direction. In this way, a negative surface charge is piled up at the end of the conductor in this direction and a corresponding positive surface charge on the opposite end. This in turn causes and electric field. After some time, you are back in a stationary state, where the force of the electric field, created through the pileup of surface charges, and the Lorentz force from the magnetic field of the electron precisely cancel each other. Through measuring the voltage drop according to this induced electric field you can (a) find out that it are indeed negative charge carriers that make the current (since the electric field would be opposite for positve charges) and (b) the density of conduction electrons.

Hi vanhees, thanks for this response.

Everything you stated I agree with, although it does not yet answer my query. I have bolded the relevant portion.

At equilibrium, there is no net force on the moving particles in the direction of the Hall voltage. Yet, on a macro scale, a wire will still bend in that direction. My confusion lies in where that net force can be accounted for. Since it is not accounted for on the moving particles, since the Hall field counter-acts the Lorentz force on them, it must be accounted for on some other particles. In this case, that must be the stationary metal cations. Since they feel no Lorentz force (they are not moving through a magnetic field), the only force they feel is from the induced Hall field. This must be the true cause for the deflection of the wire.

 

[vanhees71]

I understood that you keep the wire as a whole at rest. Of course, is will move according to the Lorentz force on the electrons, if you don't do that.

 

[uby]

I understood that you keep the wire as a whole at rest. Of course, is will move according to the Lorentz force on the electrons, if you don't do that.

No, it won't (not for the stated reason at least). And that's exactly the point of this thread. The electrons experiencing a Lorentz force also experience a force due to the induced electric field caused by charge separation which is equal but opposite in magnitude to the Lorentz force. The net force on the moving electrons in the direction perpendicular to their velocity and the applied magnetic field, after charge separation at equilibrium, is zero! The force that would cause the wire to move (or deflect, if supported) would be the force experienced by the non-moving charges due to the induced Hall field.

 

[Studiot]

Perhaps you are trying to look at this too deeply?

In order to have a magnetic field you must have a magnet of some sort.

The current in the wire produces its own magnetic field. This is a fundamental effect of current.

Thus the current affects the magnet which generates your field, exerting a force on it.

Newton's 3rd Law requires the magnet to exert an equal and opposite force on the wire.

It is likely that the magnet is moree massive than the wire and well fixed, so only the wire will appear to move as a result of this interaction.

go well

 

[uby]

Perhaps you are trying to look at this too deeply?

In order to have a magnetic field you must have a magnet of some sort.

The current in the wire produces its own magnetic field. This is a fundamental effect of current.

Thus the current affects the magnet which generates your field, exerting a force on it.

Newton's 3rd Law requires the magnet to exert an equal and opposite force on the wire.

It is likely that the magnet is moree massive than the wire and well fixed, so only the wire will appear to move as a result of this interaction.

go well

but the induced magnetic field caused by the current has zero divergence. so, wouldn't the net interaction with an applied magnetic field be zero (always positive on one side and negative on the other)?

that seems different to me than the case where a zero curl B field exerts a Lorentz force on moving charges, inducing charge separation and a counter-balancing E field which then interacts with stationary charges.

 

[vanhees71]

No, it won't (not for the stated reason at least). And that's exactly the point of this thread. The electrons experiencing a Lorentz force also experience a force due to the induced electric field caused by charge separation which is equal but opposite in magnitude to the Lorentz force. The net force on the moving electrons in the direction perpendicular to their velocity and the applied magnetic field, after charge separation at equilibrium, is zero! The force that would cause the wire to move (or deflect, if supported) would be the force experienced by the non-moving charges due to the induced Hall field.

You should do an experiment with a wire in a magnetic field. Let some current run through, and you'll see that it moves due to the Lorentz force. The motion of the positive ions of the wire can be neglected, so there's no Lorentz force excerted on them but only on the electrons. This force leads to the motion of the wire as a whole.

Of course, if you keep the wire at rest, there will be the pileup of charges at the surface as explaned in my previous posting and this leads to the Hall effect. In the stationary state, the net force on the electrons (and thus on the wire as a whole) vanishes.