Derivations of Maxwell's equations

MAXWELL'S EQUATIONS

In 1865 James Clerk Maxwell framed the theory of electromagnetic radiation through four equations and explained about the fundamental relations between electric and magnetic fields. Those four equations are called as Maxwell's equations.

Maxwell's equations are formulated based on the fundamental laws such as (i) Gauss law for electricity, (ii) Gauss law for magnetism, (iii) Faraday's law of electromagnetic induction and (iv) Ampere's law.

The formulated Maxwell's equations are,

Let us discuss the derivations for obtaining the above equations.

DERIVATIONS OF MAXWELL'S EQUATIONS

(i) Maxwell's first equation from electric Gauss law

Let us consider a dielectric medium of surface 's' bounded by the volume V. If 'Q' is the total charge in the dielectric material whose charge density is ρ then,

According to Gauss law, for electric field we can write,

Since Air is a perfect dielectric ε_r = 1 (for Air)

             ……………. (2)

Substituting equation (2) in (1) we get

Since the total charge Q is equal to the charge density ρ over the volume V, we can write

Comparing eqn. (3) and (4), we get

Equation (5) represents the Maxwell's first equation in integral form.

Differential form

The differential form of Maxwell's first equation shall be obtained from the integral form of Maxwell's first equation, by converting the surface integral into volume integral.

∴ Applying Gauss divergence theorem to LHS of eqn (5), we get

          ....... (6)

From eqn (5) and (6), we can write

            ....... (7)

Two volume integrals are equal if these integrands are equal,

∴ Equation (7) becomes

          ....... (8)

Equation (8) represents the Maxwell's first equation in differential form.

(ii) Maxwell's second equation from magnetic Gauss law

According to Gauss law for magnetic field, the net magnetic flux through any closed surface is equal to zero.

i.e, ϕ=0                ……………(9)

We know that the magnetic flux (Q) in terms of magnetic induction (B) is

              ……………(10)

Comparing equations (9) and (10) we get,

             ……………(11)

Equation (11) represents Maxwell's second equation in integral form.

Differential form

The differential form of Maxwell's second equation shall be obtained as follows.

Using Gauss divergence theorem, equation (11) can be written as

 …………(12)

Here, the surface bound volume is an arbitrary, therefore equation (12) holds good only if the integral vanishes.

 ………. (13)

Equation (13) represents the Maxwell's second equation in differential form.

(iii) Maxwell's third equation from Faraday's law

According to Faraday's law,

 ε = ‒ dϕ/dt         ……….(14)

Where ε → Electromotive force & ϕ → Magnetic flux

From eqn (29) we can see that the charge density becomes constant (or) in other words we can say that the change density is static. Thus we conclude that the Ampere's equation is valid for steady state conditions and is invalid for time varying fields.

Therefore, Maxwell assumed that equation (26) is incomplete and hence he modified Eqn. (26) by adding displacement current density J_d to equation (26). Hence equation (26) becomes

From Maxwell's first equation, we know . ∴ The above equation becomes

Equation (33) represents the Maxwell's fourth equation in differential form.

SUMMARY OF MAXWELL'S EQUATIONS IN DIFFERENTIAL AND INTEGRAL FORM

The summary of Maxwell's equations in differential & integral form, based on Guass Law (in both electric and magnetic fields), Faraday's law and Ampere's law is appended in the table below.

Law

1. Gauss law in electric field

2. Gauss law in magnetic field

3. Faraday's law

4. Ampere's law