Magnetic Forces and Work: An Explanation

[lugita15]

Since the magnetic force on a moving charged particle is qv X B, the work done by a static magnetic field is always zero.
But consider two parallel currents, both going from left to right. The second current will experience a magnetic force of attraction towards the first current. Therefore, it will move towards the first wire and work will be done on it by the magnetic field. But didn't we just say that it's impossible for a magnetic field to do work?

Any help would be greatly appreciated.
Thank You in Advance.

 

[Meir Achuz]

But didn't we just say that it's impossible for a magnetic field to do work?

But you didn't say that. You said no work on a moving CHARGED PARTICLE.
That statement does not apply to currents.

 

[Doc Al]

The electric field does the work

The magnetic field itself does no work on the moving charges, it merely deflects them. This deflection induces an electric field in the wires, which acts on the wire's positively charged lattice, and does the work in moving the wires together.

 

[ChaosUltima]

The reason the first wire is attracted to the second wire is due to the moving charges inside the wire. When a current flows in the second wire, it creates a magnetic field that goes through the first wire perpenticular to the axis of the first wire, so a charge moving in the first wire with a certain velocity v and charge q will experience a force towards the second wire (if the currents are in the same direction) or away (if the currents are in opposite directions) from the second wire by qv x B. The same goes for the second wire, when a current flows in the first, a resultant force is created on the moving charges in the second wire, so essentially the two wires move together or apart depending on the current direction.

 

[Doc Al]

The question was not why there's a force between the two wires, but what is actually doing the work--since a magnetic field does no work on moving charges. To answer that requires looking below the surface at the induced electric field.

 

[lugita15]

But you didn't say that. You said no work on a moving CHARGED PARTICLE.
That statement does not apply to currents.

Yes it does. The force on a current I in a magnetic field B is given by Il x B. The same argument can be applied.

 

[lugita15]

The magnetic field itself does no work on the moving charges, it merely deflects them. This deflection induces an electric field in the wires, which acts on the wire's positively charged lattice, and does the work in moving the wires together.

It is my understanding that a magnetic field can only induce an electric field in a loop if there is a changing magnetic flux through the loop. What loop are you considering, and how is the magnetic flux changing through the loop?

 

[Doc Al]

No loop, no changing magnetic field. (Perhaps my use of the term "induced" was a bit sloppy.) There are charge carriers--electrons--moving in the wire. The magnetic field deflects them to one side of the wire, creating a charge separation and thus an electric field.

 

[Hootenanny]

If you need a visual aid have a look at http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/hall.html" .

 

[da_willem]

Here's my 2 cents:

The magnetic field of wire 1 deflects the current in wire 2 in the direction of wire 1, simply by the Lorentz force IxB. So now the velcoity of the electrons in the wire is no longer parallel to the orginal direction of the wire, but it acquires a component in the direction of wire 1! The Lorentz force thus acquires now both a component in the direction of wire 1, responsible for the attraction between the two wires, and a component in the direction opposing the current in wire 2.

Notice two points:

1) To sustain this situation and to get the two wires moving towards each other the Lorentz force in the direction opposing the current in wire 2 must be overcome. It is this force that pushes the electrons forward, overcoming the opposing Lorentz force, that eventually does the work in moving the two wires together. Notice that this is just the electric force driving the current, generated in a battery for example.

2)In the entire process the Lorentz force is prependicular to the current, and thus does no work! Griffiths has a nice intuitive analogy: when you push a block up an inclined plane by exerting a horziontal force, it is you who does the work. The normal force no plays the role of the magnetic field in this analogy. It is always perpendicular to the ramp and the motion, so does no work. It has however a component in the horizontal direction that you have to overcome to move the block. It has also a vertical component that is responsible for the vertical movement of the block. But it is actually your horizontal force, merely deflected by the normal force in the vertical direction. The role of the normal force (and the magnetic force in your question) is one of redirecting a horizontal force in the vertical direction.

 

[Hootenanny]

Careful da_willem, the Lorentz force includes the electric force, which definitely does do work in this case. However, under no circumstances does the magnetic force do work; by definition the magnetic force is always perpendicular to motion and hence does no work.

 

[da_willem]

Careful da_willem, the Lorentz force includes the electric force, which definitely does do work in this case. However, under no circumstances does the magnetic force do work; by definition the magnetic force is always perpendicular to motion and hence does no work.

Sorry, I always, probably faulty, call only the magnetic part the 'Lorentz force'.

 

[Hootenanny]

Sorry, I always, probably faulty, call only the magnetic part the 'Lorentz force'.

No problem, I just wanted to avoid confusion for the OP

 

[lugita15]

No loop, no changing magnetic field. (Perhaps my use of the term "induced" was a bit sloppy.) There are charge carriers--electrons--moving in the wire. The magnetic field deflects them to one side of the wire, creating a charge separation and thus an electric field.

I understand your explanation. But consider two positive charges q moving parallel to each other, each with speed v. Due to the magnetic field of the first charge, the second charge is attracted to it. What is the source of the work done on each of the charges, if it is not the magnetic field?

 

[Hootenanny]

I understand your explanation. But consider two positive charges q moving parallel to each other, each with speed v. Due to the magnetic field of the first charge, the second charge is attracted to it. What is the source of the work done on each of the charges, if it is not the magnetic field?

Indeed there will be a magnetic force, but there will also be an electric force. Are the charges going to move towards one another or away from one another? Consider the relative strengths of the electric and magnetic fields; http://academic.mu.edu/phys/matthysd/web004/l0220.htm. Can the magnetic force ever be greater than (or even equal to) the electric force? And recall the defintion of work...

 

[neelakash]

Parallel current attracts/antiparallel current repels...

Are you people sure that only Electric field in Lorentz force law expression takes the responsibility?

I am thinking of the source driving the current.

 

[cesiumfrog]

This has been an interesting thread. I found it intuitively difficult to reconsile my experience of interaction between permanent magnets with the assertion that the magnetic field does no work. One question:

Does the magnetic field in the Stern-Gerlach experiment do work on an electron?

 

[da_willem]

Parallel current attracts/antiparallel current repels...

Are you people sure that only Electric field in Lorentz force law expression takes the responsibility?

I am thinking of the source driving the current.

Which is an electric field...

 

[da_willem]

This has been an interesting thread. I found it intuitively difficult to reconsile my experience of interaction between permanent magnets with the assertion that the magnetic field does no work. One question:

Does the magnetic field in the Stern-Gerlach experiment do work on an electron?

Hmm, that's a very interesting question. Classically the energy of a magnetic dipole in an magnetic field is $$-\vec{m} \cdot \vec{B}$$ so in an inhomogeneous magnetic field the force is given by

$$\vec{F}=\nabla (\vec{m} \cdot \vec{B})$$

The magnetic moment will move and it seems as though it is the magnetic field is responsible. But actually what I think will happen is that the current that produces the magnetic moment will decrease, or if it is sustained the force responsible does the work.

But now arises a problem, in case of a fundamental magnetic moment, spin, what happens? A particle carrying spin also has an energy -mB in a magnetic field. What force is now responsible for the work?

I think a that here we should change our language and no longer speak of a force, as there is no such thing in quantum mechanics. But still, what is responsible for the energy of a particle with spin in a magnetic field? I once saw a quantum mechanical derivation of this energy formula, I will look into it some further, it is indeed a very interesting subject...

 

[neelakash]

I am being more and more convinced about it.However,I am saying for a current carrying wire.You cannot blame me questioning about what is going to happen if we consider only two point charges...They constitute non-steady current ---nonsteady magnetic field and the force is a bizarre one...

Come to the problem.It's a fact that magnetic force attracts a wire towards another.Consider a moment when one wire is approaching another (horizontally) with speed u (w.r.t. my inertial frame).Let the current in this wire flow as usual with speed w (vertically).Now,EACH charge is moving as v= sqrt[w^2 +u^2] in my inertial frame.Magnetic force is not along horizontal.It tilts down.

The horizontal component is responsible for vertical motion of charges and only that...It can never influence any motion in the horizontal direction.Similarly,the downward force component is sustained by the horizontal movement of the wire.You also cannot say that this acts on the vertical motion of charges along the wire...Magnetic force never influence charges moving along its direction...

Then,who does the work?If you stop and think about it,you can see easily unless the battery would do work on the charges moving up the wire,the "attraction" what we see would not be there.Since the battery is out of the scene,it takes a time to conceive this.But note that if battery does this,you do not have to blame any unjust thing on any force (What about the electric force?That leads to repulsion!).All we need is to have a means to redirect the verical force of the battery into horizontal motion of the wire.The role of magnetic force is precisely this-simply to "redirect" the vertical force of the battery to horizontal motion of the wire.And this workdone is F_horizontal*dr.Since there is no apparently visible horizontal force apart from magnetic one,one may first think (wrongly) that magnetic force is doing the work...

How can I justify the concept of "redirecting"?I found it said by Griffiths in a similar example shown by him.He says that magnetic force redirects the efffort and leads to the physical reality.

 

[neelakash]

What is said by da_willem may be true...but that does not answer the actual question.
Let me be true to you.I did not understand how he got to the concept of magnetic dipole moment where we do not have a loop at all.
Since I do not know QM,I cannot say anything regarding this but this much I know that all sorts of magnetism do not arise out of QM.Only diamagnetism and ferromagnetism is a rigorous QM phenomenon.

 

[da_willem]

What is said by da_willem may be true...but that does not answer the actual question.
Let me be true to you.I did not understand how he got to the concept of magnetic dipole moment where we do not have a loop at all.
Since I do not know QM,I cannot say anything regarding this but this much I know that all sorts of magnetism do not arise out of QM.Only diamagnetism and ferromagnetism is a rigorous QM phenomenon.

Elementary particles have a property called spin, which follows from relativistic quantum mechanics (Dirac equation). Spin, acts in the presence of a magnetic field in the same way a current loop would.

 

[da_willem]

How can I justify the concept of "redirecting"?I found it said by Griffiths in a similar example shown by him.He says that magnetic force redirects the efffort and leads to the physical reality.

The electrons in the wire are deflected towards the other wire. They are however confined to the wire by electric forces (positive nuclei). What happens is that the entire wire moves towards the other wire, this movement however goes at the expense of the kinetic energy of the electrons. To maintain this energy the battery driving the electric current will have to do work to maintain the electron velocity and thereby the velcoity of the wire in total.

 

[neelakash]

That is good.So,everything OK?

 

[lugita15]

Indeed there will be a magnetic force, but there will also be an electric force. Are the charges going to move towards one another or away from one another? Consider the relative strengths of the electric and magnetic fields; http://academic.mu.edu/phys/matthysd/web004/l0220.htm. Can the magnetic force ever be greater than (or even equal to) the electric force? And recall the defintion of work...

The definition of work is integral of F.dr. If the magnetic force on an object is either parallel of antiparallel to the direction of motion, as it is in this case, there should be a nonzero work done by the magnetic force, and thus by the magnetic field. What is the explanation here?

 

[neelakash]

magnetic forces never do work.

 

[da_willem]

The definition of work is integral of F.dr. If the magnetic force on an object is either parallel of antiparallel to the direction of motion, as it is in this case, there should be a nonzero work done by the magnetic force, and thus by the magnetic field. What is the explanation here?

The equally charged particles will be repulsed by the electric force, which does the work that supplies this kinetic energy. This can most easily be seen in an inertial frame moving along with the particles, where the situation is the extremely simple one of two equally charged particles repulsing each other by the normal Coulomb force.

When you switch to the 'nonmoving' frame, relative to which the particles move with a certain velocity a magnetic field 'arises' in our description. Note that for velocities smaller than c, this attraction force is always smaller than the electric repulsion force.

In this stationary frame the repulsion force is thus diminished by the magnetic field, but should ofcourse yield the same acceleration. I think this has something to do with time dilation, if anyone knows I'm quite interested.

 

[neelakash]

Originally Posted by lugita15
The definition of work is integral of F.dr. If the magnetic force on an object is either parallel of antiparallel to the direction of motion, as it is in this case, there should be a nonzero work done by the magnetic force, and thus by the magnetic field. What is the explanation here?

F_mag=q(vxB)
and F.dl=q(vxB).(v dt)

 

[daniel_i_l]

I didn't have time to read past the first few threads but I think that the answer is obvious:
Let's say that one wire has a current going through it and the other one doesn't, in this case there will be no force. Only when the current is switched on in the second wire do they move towards each other. This means that the source of the work is whatever is "pumping" the electrons around in the wires. Since that wasn't very clear I'll explain a different way:
Lets say we have a satellite orbiting around Earth, it's spinning around and around but the only force acting on it is gravity which is perpendicular to the motion so it isn't doing any work! How can that be? Well in this case the easy answer is that since the satellite's energy stays constant throughout the orbit no work has to be done on it by Earth.
The same thing applies here. In order to find the source of the work done on the wires we have to look at the energy source of the system; who looses energy by the force moving the wires? The source of the current (batteries..) of course.
Am I missing something?

 

[neelakash]

Good!That is a direct proof that the battery does the work.
Here the wire gains a sidewise motion that calls for a KE.This is suppiled by the battery.

I do not understand the relevance of the analogy of the planetary motion.

 

[lugita15]

The same thing applies here. In order to find the source of the work done on the wires we have to look at the energy source of the system; who looses energy by the force moving the wires? The source of the current (batteries..) of course.
Am I missing something?

I believe you are. The cause of motion is not necessarily the force which does work. Work is done when and only when integral F.dr is 0.

 

[da_willem]

I believe you are. The cause of motion is not necessarily the force which does work. Work is done when and only when integral F.dr is 0.

I think he says you have to look where the energy dissipates, and then look at the most direct force (with F.dr nonzero) that is repsonsible.

 

[radi0headfan]

I believe you are. The cause of motion is not necessarily the force which does work. Work is done when and only when integral F.dr is 0.

In response to your original post, the force is in fact perpendicular to the motion of the charge, leading to no work. I'll do my best to try and explain why. So I'll label everything as: wire 1, charge 1(travels in wire 1), wire 2, and charge 2(travels in wire 2). You said the direction of current is from left to right so I'll use the same convention. The X's stand for the magnetic field, and for wire 1, they are going into your computer screen between the wires and out of the screen on the other side.

OOOOOOOOOOOOOOOO
------------------------ wire 1
XXXXXXXXXXXXXXXXXX

OOOOOOOOOOOOOOOOO
------------------------ wire 2
XXXXXXXXXXXXXXXXXXX

So using the right hand rule, the force due to the field of wire 1(look at the field in between the lines, and it is into the page) on charge 2( moving from left to right in wire 2) will be directed from wire 2 towards wire 1(from the bottom to the top of the screen). The same idea holds for the field from wire 2 acting upon charge 1, which is directed from wire 1 to wire 2(from the top to bottom of the screen). Ok, so we've established the directions of the forces and now need to find the work done BY EACH MAGNETIC FIELD ON ITS RESPECTIVE CHARGE. So this means the work done by the field from wire 1 on charge 2 and the work done by the field from wire 2 on charge 1. The equation for work is W = Fdcos(theta). So we're interested in the parallel component of F stands for the force on the object(Ex. force from magnetic field of Wire 2 on charge 1 that goes from bottom to top of screen). However there will never be a parallel component of force on the object because the charge is traveling from left to right, thus making the distance traversed in some infitesimal amount of time also from left to right. Now due to the force, the charge may be deflected slightly towards the other wire, but at the same time, the force from the magnetic field is also going to change respectively, and it will still be oriented 90 degrees to the charge. So at all times, the force due to the magnetic field on a moving charge will always be zero.

Hope this helps.

 

[daniel_i_l]

I think he says you have to look where the energy dissipates, and then look at the most direct force (with F.dr nonzero) that is repsonsible.

Yes, that's basically what I meant.

 

[neha_dodeja]

when we solve questions we consider the work done as ilb if the work is done by the electric field how come we have this expression??could u pls derive it for me.