James Clerk Maxwell

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# 6 Appendix: a note on displacement current density

This appendix is optional reading. It is included for the sake of comparison with other texts.

The Ampère–Maxwell law,

is sometimes expressed in the form

where Jd = ε0E/∂t is called the displacement current density. The Maxwell term is then equal to μ0Jd. Setting aside the adjective ‘displacement’ for the moment, this terminology appears to be reasonable because Equation 7.32 shows that Jd has the same units as the current density J. Regrouping and renaming terms in this way cannot affect our predictions, but it does affect the language we use to describe electromagnetism, and has provoked heated discussions between physicists.

The origins of the dispute go back to Maxwell himself, who did not know that charge is a property of particles, but thought of it as a distortion or displacement in the ether. With this background, Maxwell saw no reason to place the displacement current density on a different footing to the ordinary current density, and regarded both as contributing to a total current density (J + Jd). Although this interpretation arose from a murky understanding of the nature of charge and current, it is still in fairly common use today.

This course gives a different description, which can be traced back to Lorentz. About twenty years after Maxwell's death, Lorentz promoted the modern view that charge is carried by particles, and that currents are just flows of charged particles. Lorentz insisted that charge and current densities are only sources of electric and magnetic fields. The term ε0μ0E/∂t in the Ampère–Maxwell law is therefore regarded as part of the response of the electromagnetic field, not as one of its sources. This is why I have called it the ‘Maxwell term’ – a neutral expression which carries no implication that we are dealing with any kind of current density.

Although we cannot go into the details here, Lorentz solved Maxwell's equations to show that the values of the electric and magnetic fields at a given point and time (not just their divergences and curls) can be related to the charge and current densities throughout space. Because it takes time for information to travel from distant sources to the point at which the fields are measured, we need to know the charges and currents at times before the instant when the fields are measured. This delay emerges naturally from Lorentz's solutions to Maxwell's equations. An analogy can be drawn with throwing a stone into a pond. If you want to know about the ripples reaching the sides of the pond, you need to know about the motion of the stone at an earlier time, when it struck the water.

Things are very different in the description that treats the displacement current density as a source term. In this description, the spirit of Ampère's law is retained, while the definition of the total current density is modified. The Biot–Savart law is equivalent to Ampère's law, so this means that the Biot–Savart law can be extended to time-dependent situations provided that we use the total current density (J + Jd) to define current elements. However, when we do this, it is essential to use the present values of J and Jd – the values at the precise instant when the field is measured. No delays are involved. That is why I cannot take this description literally. Since the advent of relativity, it is much more natural to use Lorentz's description, which has all the expected delays built into it.

Having said all this, it is important to remember that we are only talking about semantics. If you hear that there is a debate about the existence of the displacement current, this will almost certainly be about the interpretation of the Ampère–Maxwell law, rather than about its validity. An analogy can be drawn with the concept of centrifugal force in mechanics. Modern textbooks describe this as the fictitious outward force you feel when you are swung in a circle, and tend to use the inward centripetal force instead. Taking a leaf from mechanics, the displacement current density might be called a fictitious current density, though I have never seen this done. No doubt, tradition and respect for Maxwell are inhibiting factors.

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