Maxwell's 4th law starts with Ampere's law. Maxwell found that to make the law complete an additional factor was required. This complete law is called the Ampere-Maxwell law. Ampere's law in its incomplete form is shown here. You should review its derivation and how it was used in an earlier section. Our present focus is completing it. The I in Ampere's law is called the conduction current, dQ dt. The electric field that produces the current is constant. Maxwell added a term that represents a varying electric field. Imagine a capacitor in a circuit. Conduction current passes through all the wires in the circuit, but no current passes between the two plates. Between the two plates of the capacitor, the electric field increases in strength. The problem is that though current flows to and from the capacitor, it doesn't flow between the plates of the capacitor. Recall how you used your right hand to find the direction of a magnetic field around a current carrying wire. If you point your right thumb in the direction of the current, your fingers will curl in the direction of the magnetic field. In this figure, a hemispherical surface surrounds the positive plate of the capacitor. Its flat surface shows the expected magnetic field. Gauss' law holds for the flat side. But the curved side obviously has no associated magnetic field. There seem to be two different magnetic fields, which is a contradiction. Maxwell resolved this contradiction by creating something called a displacement current. The displacement current is equal to the permittivity of free space times the rate of change of electric flux with respect to time. So, finally, here it is, Maxwell's 4th law, also known as the Ampere-Maxwell law. It is the most significant of all of Maxell's 4 laws. It says that a magnetic field is produced by a varying electric field and also by a current.

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