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.