As we turn on a current

Maxwell's equations presented the laws of electromagnetism in quantitative form, and made it possible to use the powerful techniques of mathematics to draw logical conclusions based on the combination of diverse experimental data. Maxwell showed that the solution of the equations predicted that an electromagnetic signal, once initiated, will propagate at a certain fixed velocity.

Rather than solving the equations formally, it is perhaps more instructive to consider each equation in turn, and show that a remarkable chain of events is set in motion by a simple act such as switching on an electric current.

Figure 10.19 Magnetic field around a current-carrying wire.

As soon as we turn on the current a magnetic field (B) appears which encircles the wire, anti-clockwise as we look at it from above. The diagram depicts the situation where we have a steady current i flowing in the wire. If we go back to the moment when the current was switched on, there was at first no current and no magnetic field. Then, as the current started to flow, there was still no magnetic field at points removed from the wire. We can deduce from Faraday's law that the magnetic field cannot appear instantly. If it did, there would be an instantaneous change of the magnetic field from zero to its final value everywhere, resulting in an infinitely large rate of change of flux, d(B/dt, in all areas of the surrounding space, which in turn would induce an infinitely large electric field E. We have no option but to conclude that the creation of the B field can only spread out from the wire at a finite speed (v).

To make the discussion simpler, let us assume that the magnetic field created by the current is uniform. (Such a uniform B field would result if we had an infinitely large moving sheet of charge instead of the electric wire.) The rate of propagation of the magnetic field is the same in the two cases.

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