Topic : Maxwell’s Equation in Isotropic Dielectric Medium

Maxwell’s Equations

Maxwell in 1862 formulated the basic laws of electromagnetic in the form of four fundamental equations. These equations are known as maxwell’s electromagnetic equations. These equations are based upon the well known laws such as gauss’s law of electrostatic, gauss’s law of magnetostatic, faraday’s law of electromagnetic induction and ampere’s circuital law. The integral forms of these equations are given below : The total flux coming out of a closed surface is equal to the net charge enclosed The surface integral of magnetic flux density over a closed surface is zero.

The net emf induced in a closed path is equal to the surface integral of negative time rate of change of flux density over the surface bounded by the closed path. The total emf around any closed path must be equal to the surface integral of conduction and displacement densities over the surface bounded by the closed path.

The differential forms of these equations are given below : Faraday’s Law of induction Ampere’s Law Gauss’s Law for electricity Gauss’s Law for magnetism

Derivation of Maxwell’s Equations in Differential Form Maxwell’s first equation According to Gauss’s law of electrostatic, the total electric flux (mathematically, the flux passing through an area element of surface) is equal to the scalar product of electric field E at that element and vector area dS, i.e., E∙𝑑𝑆. For any medium, 𝑠 𝐸∙𝑑𝑆= 𝑞 ∈ …(1) If charge q inside the close surface is continuously distributed and 𝜌 is the volume charge density, then 𝑞= 𝑣 𝜌𝑑𝑉 …(2)

From eqs. (1) and (2), we get 𝑠 𝐸∙𝑑𝑆= 1 ∈ 𝑣 𝜌𝑑𝑉 …(3) This is integral form of Maxwell’s first equation. Using Gauss’s divergence theorem, 𝑠 𝐸∙𝑑𝑆= 𝑣 𝛻 ∙𝐸 𝑑𝑉 …(4) From eqs. (3) and (4), we get 𝑣 𝛻 ∙𝐸 𝑑𝑉 = 1 ∈ 𝑣 𝜌𝑑𝑉 𝛻 ∙𝐸 = 𝜌 ∈ …(5)

Maxwell’s Equations in Isotropic Dielectric Medium A dielectric medium is one in which the conduction current is almost zero in comparison to displacement current. Such a medium is treated as perfect dielectric or loss less medium. In such a case, the wave equations become : 𝛻 2 𝐸= 𝜇∈ 𝜕 2 𝐸 𝜕𝑡 2 𝛻 2 𝐻= 𝜇∈ 𝜕 2 𝐻 𝜕𝑡 2 𝛻 2 𝐷= 𝜇∈ 𝜕 2 𝐷 𝜕𝑡 2 𝛻 2 𝐵= 𝜇∈ 𝜕 2 𝐵 𝜕𝑡 2