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Ampere’s circuital law

According to this law, the line integral of magnetic field along any closed path or circuit is μ0 times the total current threading the closed circuit, i.e., 78454.png.

Inconsistency of Ampere’s law

Maxwell explained that Ampere’s law is valid only for steady current or when the electric field does not change with time. To see this inconsistency consider a parallel plate capacitor being charged by a battery. During the charging time varying current flows through connecting wires.
Fig. 3
Applying Ampere’s law for loop l1 and l2,
But 78470.png (Since no current flows through the region between the plates). But practically it is observed that there is a magnetic field between the plates. Hence, Ampere’s law fails
i.e. 78476.png.

Modified Ampere’s circuital law or Ampere- Maxwell’s circuital law

Maxwell assumed that some sort of current must be flowing between the capacitor plates during charging process. He named it displacement current. Hence modified law is as follows
or 78488.png
where ic = conduction current = current due to flow of charges in a conductor and id = Displacement current = 78503.png = current due to the changing electric field between the plates of the capacitor.

Maxwell’s equations

  • 78520.png (Gauss’s law in electrostatics)
  • 78531.png (Gauss’s law in magnetism)
  • 78537.png (Faraday’s law of EMI)
  • 78546.png (Maxwell-Ampere’s circuital law)

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