Maxwell’s Equations: Deferential and integral from of Maxwell Equations

What are the Maxwell’s Equations? Derive Deferential and integral from of Maxwell’s Equations.

Maxwell`s equations are a set of four equations which tells us about the production, propagation and interaction of electric and magnetic fields. These are the most fundamental equations of electricity and magnetism. Maxwell picks four basic laws of electricity and magnetism, viz. Gauss` law for electric field, Gauss` law for magnetic field, Faraday`s law of electromagnetic induction and Ampere`s law. After correction in Ampere`s law by introducing the concept of displacement current, Maxwell combined all these laws into a set of four equations known as Maxwell`s equations.

The Maxwell`s equations in free space (or vacuum) are:

  1. Gauss`s law for electric field

  2. Gauss`s law for magnetic field

  3. Faraday`s law

  4. Ampere`s law (with Maxwell`s correction)

Derivation of Maxwell`s Equations:

(i) Gauss` law for electric field:

Let us consider a charge Q is enclosed by a surface S the volume of which is V. Then according to Gauss`s law

(ii) Gauss` law for magnetic field:

As we know that, an isolated magnetic pole does not exist, therefore the magnetic lines of force are the closed curves.  Thus, the number of magnetic lines of force that enter a closed surface present in a magnetic field is same as the number of magnetic lines of force leaving that surface. That is the net magnetic flux through a closed surface is always zero. i.e.

(iii) Faraday`s law:

According to the Faraday`s law the induced emf in a closed loop is

(iii) Ampere`s law (with Maxwell`s correction):

As we know that, a moving charge (or current) produces a magnetic field. Ampere derives a relation between current and magnetic field produced by it which is called as Ampere`s law. According to Ampere`s law “the line integral of magnetic field along a closed curve C is equal to Mu-Zero times the net current I flowing through the area enclosed by the curve”. i.e.

The above equation represents the differential form of Ampere`s law which is applicable only for the steady current. When we take divergence of above equation then it seems to be inconsistent for non-steady current as

Maxwell`s equations in an isotropic dielectric medium


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