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The basic equations of electromagnetism are derived from the Gauss’s law of electricity and Gauss’s law of magnetism, Ampere’s law and Farraday’s law. Consider any closed surface in a region such that Gauss’s law can be applied to the electric as well as magnetic fields:
Since the surface encloses no charge the line integrals are zero for both the fields. This implies an important symmetry between electric and magnetic fields.
Now let us consider any closed path in this region and apply Farraday’s law and Ampere’s law as follows:
From these equations it is realized that the symmetry observed in the application of Gauss’s law no longer exists in the application of Farraday’s and Ampere’s laws. It is because the second sets up equations imply that a varying magnetic field can set up an electric field but not the vice versa. This problem was solved by considering a displacement current.
Induced Magnetic Field and the Displacement Current:
If B is the induced magnetic field due to a symmetric current carrying conductor then the line integral of the induced magnetic field is equal to μ0I
i.e.
This is the form of Ampere’s law which is valid if the electric fields present are constant in time. Now, for discontinuity of current occurs the above integral vanishes. However, this is a problem if it vanished and can be overcome by considering another current called displacement current defined as:
Then the line integral of the induced magnetic field is given as:
Hence, the magnetic fields are produced both by conduction currents and by changing electric fields.
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