Summary of Electromagnetic Forces

OK, so what is my point in the preceding derivations?

1)  We assumed only the 1/r electrostatic potential, the Special theory of Relativity and a few very basic ideas from Quantum Mechanics.  From this we showed that the electromagnetic forces follow from these three ideas.  In particular I think these derivations show how a relatively simple force between stationary particles can be transformed by Special Relativity into a rather complex set of forces (Electromagnetism).  I think this is an important fundamental concept.  It shows a deep relationship between Electromagnetism, Relativity and Quantum Mechanics that is not taught in books on these subjects.

2)  Note that in deriving how to use the 4-vector potential to calculate forces, we had to assume that the energy of interaction between the source and test charge was located at the source charge.  This is in direct conflict with the conventional understanding that the energy is in the electromagnetic field in the space between and around the particles.  I think this may be a very important result.  In fact, it challenges some very basic ideas in physics, such as that the energy of the interaction propagates through space from the source to the test charge.

I recently read "The Maxwellians" by Bruce Hunt (see http://www.amazon.com/Maxwellians-Cornell-History-Science/dp/0801482348).  This excellent book tells the history of electromagnetism, and the people who developed the theory in the 1880's and 1890's after Maxwell.  I did not know until reading this book that Maxwell never wrote the equations that carry his name.  It was Heaviside that took Maxwell's 20 field equations and distilled them down to 4 vector differential equations.  Maxwell's original equations included a combination of potentials and fields apparently.  Heaviside's reworking of them emphasized the fields and almost completely suppressed the potentials.  Heaviside was also one of the promoters of the idea that the energy is carried in the electric and magnetic fields, like we have all been taught.  As a result of the derivations just posted I am starting to question this dogma.

There is no doubt that solving electromagnetic problems using E and B fields (and D and H) is tremendously convenient.  The power of this method is that through the use of some simple boundary conditions, one can almost completely ignore the presence of charges on metal and in dielectric materials.  The alternative, to integrate the potentials from all these charges, requires that you first solve for the motions of all the charges acted on by whatever is driving the system (i.e. the radio transmitter, or light source, etc.).  This presents a bit of a chicken and egg problem.  How can you find the fields without knowing the motions of the charges?  How can you find the motion of the charges without knowing the fields?

In the physical world this is not a problem because each charge is affected only by charges on its past light cone.  And the charges on the past light cone are not affected by charges on their future light cone.  (Actually, in a future post I will argue that this is not completely true.)  In principle this problem can be solved, but it can be very difficult mathematically without using Maxwell's equations for figuring out the fields first.  As a practical tool Maxwell's equations are brilliant.

But I wonder if they are the correct physical model of the world.  In the preceding posts I've shown that the forces on charges can be calculated without referring to electric or magnetic fields, but only using potentials.  What is the real, physical, thing, the potentials or the fields?  This may be a pointless philosophical question if they both give the same results.  But it appears to me that they might not.  The potential approach requires the energy of interaction to be located on the source charge.  The potential approach also subtly implies action at a distance.

In fact, I learned in Hunt's book that there was a great debate about this very subject in the early 1890's at a conference in England.  It is sometimes called the "murder of phi".  One argument against the potentials was that they imply action at a distance, which was strongly disliked.  The result was that everyone agreed that it was the fields that are real and that the potentials are mathematical fictions that are useful in some problems, but are not real.  This was Heaviside's position apparently.  But this was all done before Relativity and Quantum Mechanics.  Relativity is all done with potentials.  Relativistic electrodynamics does use the electromagnetic field tensor, consisting of components of the E and B fields.  But it is defined in terms of derivatives of the 4-potential.

Quantum Mechanics almost demands action at a distance.  Wouldn't action at a distance solve the issues with quantum entanglement and EPR experiments?  Of course that would violate Special Relativity unless the interaction is on the light cone, but of course it is!  The Aharonov-Bohm effect (http://en.wikipedia.org/wiki/Aharonov-Bohm) strongly implies that the potentials are real.  This effect  demonstrates that there are observable changes to the phase of a charged particle's wave function due to the potential, even when there are no gradients in the potential to cause an "electric" or "magnetic" field.

If the potentials are real, and not just mathematical conveniences, then it seems to me that that also means that the energy of interaction is NOT located in the space between the charges, but rather at the charges themselves.  That would be a significant development in our understanding.  I probably need to do more to justify this claim, however, if I'm going to convince people that this is correct.

In future posts I will derive "electromagnetic" radiation from an oscillating electric dipole using potentials, and get the classical result.  I will also calculate radiation resistance without using Poyntings Theorem, but only potential interactions between the radiating charges and charges absorbing the "radiation".  I also plan to show that blackbody radiation can be derived using potentials and NOT using cavity modes.  (I'm still working on that, but I think I see how to do it.)  Finally, I have some posts on General Relativity that fit in with all this and give some insight into how the force of gravity is fundamentally different from electromagnetism.

I would very much like to discuss these ideas with anyone interested.  Especially if you don't buy into my arguments!

Magnetic Forces Part 3

Magnetic Forces: Part III, Motion Perpendicular to Current (via 4-potential)

Now consider the same current carrying wire as in the previous section, but this time with the test charge moving towards the wire.  Once again we will assume a wire aligned with the y axis that crosses the X axis at x = x0, and a test charge at the coordinate origin (i.e. a distance r= x0 away from the wire.  This time the test charge is moving towards the wire with speed beta_OT in the +x direction.

The wire and its currents are the same as for the last section, so the 4-potential in the observers frame is again

Transforming this to the test charge frame is a boost in the x direction.  Since the charge line is perpendicular to the direction of boost, the vector potential is only multiplied by  gamma_OT, which gives

From this we calculate the force again as

Now we need to be careful.  Since the charge is moving towards the wire, the vector potential is changing with time, so we get:

We need to be careful here too to recognize that x and dx are in the test charge frame coordinates, and that we will need to transform these to the observers frame in the next step.  Likewise the time dt is in this frame, and t will need to be transformed as well.  It turns out that the dx factors cancel out, however, and we can just do the usual transformation of the force to get the force in the observers frame:

This can be generalized for all components of the velocity and the vector potential, when the test charge is moving perpendicular to the vector potentials:

Total Magnetic Force Around a Wire

The results of the last two sections can be combined to give the total force due to a particle moving in a magnetic field.  We get:

Which can now be recognized as

OK, maybe that isn't obvious, but if you work it out, you will find that these are the same thing.

In terms of the magnetic fields this is:

Magnetic Forces: Part 2 - Parallel Motion via 4-Potential

Magnetic Forces: Part I, motion parallel to current (via the 4-potential)

We will now derive the previous result using the 4-potential.  This proves to be much simpler algebraically, and after the previous derivation we can see how the 4-potential calculation reflects the physics revealed in the previous derivation.  The problem set up is exactly the same as in the last post.

This time we calculate the 4-vector potential for both the stationary positive charges and the moving negative charges.  In the observers frame we have:

where X is the length of the wire, assumed very long compared to x.  Since the positive and negative charge densities are equal in this frame, this can be written

Where

This potential transforms as a 4-vector, so in the test charge frame the Lorentz transform gives

The test charge will now respond to the gradient of these potentials as derived previously, resulting in the force:

The test charge is not moving towards or away from the wire in this case, so the time derivatives are all zero and thus the time component of the force (the rate of change of energy) is zero.  Likewise the time derivative of the vector potential in the spatial term is also zero.  This leaves the gradient term:

The only direction with a gradient is towards/away from the wire, which is the x direction, so we get:

Transforming this to the observers frame gives (where r is the radius from the wire)

Which as before can be related to the currents and normal velocities

This is the same result obtained by the physical argument in the last section.  This can also be written in the observers frame as:

This result can be generalized for any motion of a test charge parallel to the vector potential, or to a component of the vector potential:

Note that this is not the complete magnetic force yet.  We still need to calculate the force for a charge moving towards or away from the wire.  We will do that in the next post.

Magnetic Forces: Part 1

Magnetic Forces: Part I, motion parallel to current (via charge density)

We now consider the forces on a test charge moving parallel to a current carrying wire.  The figure below illustrates the situation from the point of view of the observer.  The wire is represented by the blue dots and red dots with vectors on the right side.  The blue dots are the stationary positive charges, and the red dots with vectors represents the moving electrons carrying the current.  The green dot represents the test charge moving parallel to the current carrying wire.  The charges and charge densities are shown in the appropriate colors, with subscripts appropriate to the observers frame.

We are assuming that the net linear charge density on the wire is zero in the observers frame:

Since the charges are arranged in a line in the direction of motion, we can calculate the charge density in the source charge frames and write:

Where rho_S+ and rho_S- are the net positive linear charge density (nuclei and bound electrons) and net negative linear charge density (conduction electrons) in their respective rest frames and gamma_OS+ and gamma_OS+ are the Lorentz transform coefficients that give the apparent charge density in the observers frame.  This gives the effect of a positive current moving in the -Y direction.  Therefore we have:

This allows us to express the charge densities in the rest frame of each charge type as

and

Now consider a test charge traveling parallel to the wire in the +y direction a distance r=x₀ from the wire and at speed beta_OT.  The positive and negative charge densities are altered by the Lorentz transform.  Because the Lorentz transform is not linear in beta, we need to work with the charge densities in their rest frame and transform them directly to the test charge frame.  We need to use the Relativistic addition of velocities formula to calculate the beta's correctly, giving the net charge on the wire as seen by the test charge in its rest frame:

Where we have assumed that the positive charges are stationary in the observers frame.  We now use the expression for rho_S- derived above and factor out the rho_O+:

Amazingly, the quantity in parentheses reduces to just

which we now show.  First, factor out the denominator in the lower right radical:

Expand the remaining terms in that radical and refactor:

Now one of the denominator radicals cancels the one in the numerator, and the two terms now have same denominator:

The denominator is recognized as gamma_OT:

The expression -rho_O+ beta_OS- is just the current in the wire, I, as seen in the observers frame (divided by c).  Thus we see that the net charge on the wire seen by the test charge is proportional to the current and also the factor beta_OT, the speed of the test charge.

The force on the test charge can now be calculated.  In this frame the test charge is stationary so we don’t need to worry about magnetic effects.  From the result derived a couple of posts ago we have:

In the test charge frame, since it is moving parallel to the wire, the time derivatives are both zero.  Substituting the expression for the potential around a wire of length 2X (where X >>x) gives

and taking the derivative wrt x for x<x_0 gives:

And putting in the expressions for the charge density, and changing x to r gives:

We now want to know what is the force seen in the observers frame.  The force 4-vector does not exactly transform under the Lorentz transform, but the energy-momentum, dp, does.  Since this energy-momentum is perpendicular to the direction of the motion of the test charge it is unchanged however.  The dt in the denominator transforms to gamma_OT dt in the observers frame, so the transformed force is:

Which gives the conventional force in the observers frame:

The expression rho_O+ beta_OS- is just the current as seen in the observers frame (divided by c), and , of the test charge (divided by c), so we can write this as:

This force is directly towards/away from the wire.  If I_x is positive (i.e. a positive current flowing in the +y direction), and q_t is positive, then this force is in the minus r direction, i.e. towards the wire.  Comparing this with the Lorentz force due to the magnetic field around the wire:

From this we can then write:

Which of course is a well-known result from classical electromagnetism.

This derivation is pretty complex conceptually and algebraically.  What it shows however is that what we call the magnetic force, for a charge moving parallel to a current carrying wire, is the result of the change in charge density due to the Lorentz transformation to the test charge frame.  The test charge is merely responding to the electric field that it sees on the wire in the test charge frame.  

This very clearly shows that the magnetic force is a relativistic effect, at least for the case of motion parallel to the current.  In the next post we will show that motion towards or away from the current also results in what we call the magnetic force.  And again it will be seen that the test charge is just responding to, in that case, the rate of change of the vector potential.

Magnetic Forces: Intro

Magnetic Forces: Introduction

We now consider the forces on moving test charges due to currents.  What we normally mean by a current is a wire that has charges moving along its length, and counter charges that are stationary and balance most, if not all, the electrostatic field from the moving charges.  In a real wire with current, there is a stationary matrix of positive charges and a moving fluid of negative charges.  The positive charges are the nuclei of all the atoms of the conductor, reduced by the negatively charged bound electrons.  The moving negative charge is the collective motion of all the conduction band electrons.  To a very good approximation, the positive and negative charge densities sum to zero in the observers frame. 

In the calculations that follow, we will be dealing with three different reference frames: the source charge rest frame, the test charge rest frame, and the observers frame.  There are two source charge frames in this problem, one for the (stationary) positive charges and another for the moving negative charges.  The test charge, whose motion we are trying to understand, is generally moving with respect to one or more of the source charges.  We have already calculated in the earlier sections the forces on a stationary test charge.  We will build on that understanding to calculate how the test charge moves in the frame of an observer  that is not at rest with the test charge.

With three different reference frames, connected by the Lorentz Transformation, it can get very confusing which frame which parameter is based in.  To try to keep this all straight we will follow a strict nomenclature of subscripts:

-All parameters in the source charge frame will have a capital S subscript: e.g. x_s, rho_x etc.  These subscripts may in addition have a + or – to indicate the polarity of the source charge.

- All parameters in the test charge frame will have a capital T subscript: e.g. x_T, rho_T etc.

- All parameters in the observers frame will have a capital O subscript: e.g. x_O, rho_O etc.

-In addition, all parameters that relate to motion between frames will have two subscripts indicating first the reference frame in which the parameter is observed, and second the reference frame it refers to.  For example, the parmeter gamma_OT is the Lorentz Transform gamma parameter used to transform from the observers coordinates to the test charge coordinates.  Likewise beta_TS is the velocity of the source charge as seen from the test charge frame.

We can now construct the picture of this problem in these three frames, with all the parameters we will need defined in each frame:

Observers Frame:  We will place the origin of all the frames at the test charge.  In the observers frame the wire with the current is parallel to the Y axis and crosses the X axis at position x_O and z_O = 0.  Thus the test charge is momentarily at a radius r_O = x_O from the wire. 

We will be considering two cases: one with the test charge moving along the Y axis, parallel to the wire, and another case where the test charge is moving toward/away from the wire.  We will also briefly consider the test charge moving in the Z direction, just to show that there is no magnetic force resulting from that motion.  The velocity of the test charge is called beta_OT in this frame, and has a corresponding gamma_OT.

The source charges have velocities beta_OS+ and beta_OS-, with associated gamma parameters gamma_OS+ and gamma_OS+.  Likewise the ordinary velocity of the test charge is v_OT.

Since the positive charges are stationary in the observers frame, the density of positive charges in their own frame is the same as that in the observers frame:

The negative charges are moving, but as seen in this frame have the same density as the positive charges, but the opposite sign:

The test charge we will simply call q_T, and is the same in all frames.

Source Charge Frame:  There are two source charge frames, one for the positive charges and one for the negative charges.  For simplicity, and to understand the most typical situations, we will assume the positive charges are stationary in the observers frame.  The negative charges are moving along a line parallel to the Y axis.  In the negative source charge frame the observer frame is moving with a velocity:

And

The density of positive charges in their own frame is the same as that in the observers frame:

The negative charges have a density in the observers frame of  which we assume is equal to the positive charge density:

The test charge is also moving at a velocity  with Lorentz Transform factor  that we will calculate for each case.

Test Charge Frame:  From the test charge frame, the velocity of the observer is:

And

The charge densities of the positive and negative source charges are, in general, different from the densities in the observers frame.  We will have to calculate these for each problem.  We will call them rho_T+ and rho_T-.  The motion of the source charges also will have to be calculated for each problem.  They will be called beta_TS+ and beta_TS-.  The corresponding gamma parameters are gamma_TS+ and gamma_TS-.

One final introductory note concerning the speeds of the charges in a current carrying wire:  The density of charges in typical wires (e.g. copper) is extremely high, so even at very high currents the charges are actually moving quite slowly.  For example in copper conducting current at the maximum limit that copper is capable of maintaining continuously, the speed of the electrons is about 1 mm/second.  This shows that normal magnetic fields are the result of charges moving at non-relativistic speeds.  In our calculations we will not restrict ourselves to only non-relativistic moving charges.  Our result will be completely general.