The relativistic nature behind magnetic forces is well understood. It is commonly demonstrated via thought experiment by imagining a free charge in be... Magnetic forces - Capacitors - Lorentz covariance - Electromagnetic forces - capacitors - magnetic forces - magnetism - magnetic forces - capacitor plates - Coulomb law - Biot-Savart law - Lorentz force - applied electromagnetics - Relativistic effects - magnetism - electromagnetic field - electromagnetic forces

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James Nagel

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Abstract The relativistic nature behind magnetic forces is well understood. It is commonly demonstrated via thought experiment by imagining a free charge in between a pair of moving capacitor plates. This paper presents a greatly simplified version of the same thought experiment, by deriving equivalent results from a system of two moving point charges. The derivation only requires knowledge of Coulomb's law, the Biot-Savart law, and the Lorentz force, and can therefore be readily presented to an introductory-level course in applied electromagnetics. Keywords: Relativistic effects; magnetism; electromagnetic field; electromagnetic forces

A

lthough the magnetic interaction between charged particles is well known to be a physical manifestation of special relativity, modem introductory-level textbooks tend to neglect this very interesting fact when covering magnetism. As a result, students are frequently left with a limited physical understanding of the magnetic force, even though a few simple thought experiments can clearly demonstrate its origins.

One of the more common thought experiments is to imagine a test charge in the presence of a moving parallel-plate capacitor, which was well explained in [1]. Although this scenario works well to demonstrate the relativistic nature behind magnetism, the same concept may be more fundamentally demonstrated in an introductory-level course by considering the case of only two moving point charges. Out of this very simple system, it is possible to derive a purely relativistic expression by using nothing but Coulomb's law, the Biot-Savart law, and the Lorentz force. The goal of this paper is thus to introduce a more fundamental approach to demonstrating the physical origins of magnetism. The derivation is intended for an undergraduate course in applied electromagnetics, where the Lorentz force and the Biot-Savart law are typically given as the fundamental expressions behind magnetic interactions. To begin, consider the system in Figure 1, which depicts two equal charges (labeled a and b) moving parallel to each other with velocity v = VZ, and separated by some distance r. Naturally, there exists a Lorentz force, F, between the two particles. This is simply defined by the superposition of an electrical force, Fe' and a magnetic force, Fm : (1)

(2)

The magnetic component is found by solving for the magnetic force acting on particle b. For a moving charge in the presence of a magnetic field B, this is simply given by

Fm=qvxB.

(3)

Note that the magnetic field through which particle b moves is induced by the motion of particle a, and the strength of that field follows the Biot-Savart law. In most introductory-level courses, this is expressed in terms of infinitesimal currents, and is given as B = Pol dIxr 41ir 2

(4)

'

L.l! v

v

q

q r

The electrical component is readily found by applying Coulomb's law, which tells us the electrical repulsion on particle b due to the presence of particle a. This is expressed as

Figure 1. Two equal charges traveling parallel to each other with some velocity v.

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IEEE Antennas and Propagation Magazine, Vol. 51, No.1, February 2009

where I is the current's magnitude, dl is the differential length vector that points in the direction of current flow, and ~ = vector that points from charge a to charge b.

y is the unit

To gain more insight into this question, consider the ratio of the magnetic-force magnitude to that of the electrical force, which is readily calculated to be

For the case of simple point charges, the Biot-Savart law can be rewritten in a more meaningful form by first noting that Idl = dq dl dt 1

= dq dl . 1 dt

(5)

The expression dll dt represents the velocity of some moving piece of differential charge dq, and can therefore be replaced with the

velocity vector v = dli dt = vi. For the case of a single point charge, the differential charge dq contains the total charge at that point, and we may simply replace dq = q. Using these substitutions, we are left with [di =qv.

(10)

(6)

where c = 1/~JloEo is the speed of light in a vacuum. From this expression, we see that as the velocity of the reference frame approaches the speed of light, Fm ~ -Fe' and the magnetic force perfectly cancels out the electrical force! Although this makes no physical sense from a purely classical point of view, it makes perfect sense when viewed through the lens of special relativity. To see why, observe what happens when the electrical component is factored out from the Lorentz force:

Thus, for the case of a moving point charge, the Biot-Savart law may be expressed as

(11)

(7)

If Equation (7) is substituted back into Equation (3), we are then given the total magnetic force acting on particle b, which is induced by the motion of particle a: (8)

Finally, this expression can be plugged back into Equation (1) to find the total Lorentz force acting on particle b: (9)

This expression is equivalent to [1, Equation (28)], where it is shown to be equivalent to a Lorentz transformation acting on the Coulomb force alone. Thus as this simple thought experiment clearly demonstrates, the magnetic force is not really a force at all, but rather a physical manifestation of special relativity. The advantage to this approach is that the system consists of only two moving point charges. Such a system is greatly simplified over the moving parallel-plate capacitor system from [1], and avoids the complication of any surface currents or charge densities. It also has the advantage of emphasizing Equation (7) as the pointform of the Biot-Savart law, which is useful for students to understand as a more fundamental form of Equation (4) that links relativity to magnetism.

By symmetry, the force on particle a is then simply the negative of this value.

References

As we can plainly see, the magnetic force acts in the opposite direction to the electrical force, and varies directly with the square of the velocity of the two particles. Intuitively, this result should feel very paradoxical, because the two charges exist in the exact same frame of reference. Why then should the force between two charges vary, simply because we, the observers, are moving relative to them?

1. J. W. Arthur, "Fundamentals of Electromagnetic Theory Revisited," IEEE Antennas and Propagation Magazine, 50, 1, February 2008,pp. 19-65.~

IEEE Antennas and Propagation Magazine, Vol. 51, No.1, February 2009

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