Maxwell’s Fourth Equation
Maxwell’s fourth equation is derived from Ampere’s Law, which states that “the magnetic field divergence is always zero.” The expression for Maxwell’s first equation can be expressed mathematically as:
[Tex] ▽× H = J + (\frac{∂D}{ ∂t} ) [/Tex]
Derivation for Maxwell’s fourth Equation
According to Ampere’s circuital law
∮ B.dl = μ0i —– (1)
According to Stoke’s theorem–
∮ B.dl = ∮S ( ∇ × B ).ds —– (2)
From equation (1) and equation(2)
∮S ( ∇ × B ).ds = μ0i —– (3)
where i = ∮S J . ds —– (4)
So from equation (3) and equation (4)
∮S ( ∇ × B ) .ds = μ0 ∮S (J . ds )
∮S ( ∇ × B ) .ds – μ0 ∮S (J . ds ) = 0
∮S [ ( ∇ × B ) – μ0 J ] .ds = 0
( ∇ × B ) − μ0 J = 0
( ∇ × B ) = μ0 J
As we know that B = μ0 H
∇ × H = J
Modified Maxwell’s Fourth Equation
The modified Maxwell’s fourth equation is the differential form of the modified Ampere’s circuital law.
We know the modified Ampere’s circuital law-
∮ B . dl = μ0 i + id
Where id = Displacement current
Therefore the modified Maxwell’s fourth equation can be written as-
∇ × H = J + Jd —– (1)
Where Jd = Displacement current density
And its value of Jd is :
Jd = ϵ0 (∂E/∂t)
and
Jd = ∂D/∂t ( ∵ D = ϵ0E)
Now substitute the value of Jd in equation (1)
Therefore , ▽× H = J + (∂D / ∂t ) .
This is the required equation
Maxwell’s Equation
Maxwell’s equations are like the instruction manual for how electricity and magnetism work. They were created by a smart scientist named James Clerk Maxwell in the 1800s. Since these equations help us understand everything from how lights work to how our gadgets and technology function, they are extremely significant. In this article, we’ll see Maxwell’s Equations in detail, in which there are four equations that forms the description of the topic.
Table of Content
- Maxwell’s Equations
- Gauss’s Law
- Maxwell First Equation
- Gauss’s Law for Magnetism
- Maxwell’s Second Equation
- Faraday’s Laws of Electromagnetic Induction
- Maxwell’s Third Equation
- Ampere’s Law
- Maxwell’s Fourth Equation