An important extension to Amperes law
Maxwell soon realised that one of the laws, namely Ampere's law, as given by equation (10.1), is incomplete. This equation correctly quantifies the magnetic field produced by an electric current, i.e. by moving electric charges. Maxwell pointed out that it is also possible to have a magnetic field without physically moving electric charges. We can most easily understand why this is so by looking at what happens during the process of charging an electrical capacitor:
A capacitor is a device which stores charge. In its simplest form it consists of two parallel plates separated by a certain distance. Figure 10.18 depicts an electric circuit consisting of a battery, a switch and a capacitor. When the switch is closed a current flows, but the electric charges cannot cross the gap between the capacitor plates. Positive and negative charge build
up on opposite plates of the capacitor until the capacitor is fully charged.
While the current is flowing it produces a magnetic field, with magnetic lines of force encircling the wire according to Ampere's law. But are there any magnetic effects around the region where there is no actual flow of charge, i.e. around the central part of the capacitor? We can investigate this experimentally with a compass needle, but we can also make a logical deduction. The argument which forms the basis of Ampere's law does not depend on the size or shape of the loop. If an Amperian loop is drawn arbitrarily around the circuit, the integral of the magnetic field around the loop (recall IB • ds = /i0i) is equal to the current passing through any surface bounded by the loop. We could choose a surface which intersects the currentcarrying wire, or a surface within the capacitor, where there is no current. It follows that whatever is happening within the capacitor must equally contribute to the making of the magnetic field.
Maxwell noted that while there is no current through the capacitor, there is a changing electric field E. This results in a changing electric flux (pE through any Amperian loop around the capacitor. He proposed that this changing electric flux had the same magnetic effect as the flow of electric charge, and generalised Ampere's law by adding a second term of the form £0dq)E /dt to the righthand side of the equation. He called this term the displacement current (id). This is not a true current in the sense that there is no flow of charge, but has the same effect (and the same dimensions) as a physical current. The generalised form of Ampere's law now becomes:
The value of the integral on the LHS is independent of the location of the Amperian loop and equals ¡i0 times the sum of the charge current plus displacement current threading the loop at that instant.
Maxwell's extension does much more than add an extra term to a certain equation. It states that a magnetic field can be produced in the absence of moving charges. It can also be created by an electric field which is changing with time. The charges which originally created the electric field no longer play a role in the process, which continues without them. The fields take on a new reality.
Maxwell's fundamental work Treatise on Electricity and Magnetism was first published in 1873 and can be ranked alongside Newton's Principia as one of the cornerstones of a new scientific era.
10.6.3 The four laws 1. Coulomb's law/Gauss's theorem
The electric flux through any closed surface = net charge inside.
2. Magnetic lines of force always form closed loops
There are no single magnetic poles. Expressed in Gaussian form
Magnetic flux through any closed surface = zero.
3. Ampere's law (plus Maxwell's extension)
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