Ampere’s Circuital law; This law states that the circulation of magnetic field intensity around any closed path is equal to the sum of free current and displacement current flowing through the surface bounded by path.
the term `\frac{dD}{dt}` is displacement current density.
Maxwell’s Equation-Modified Ampere’s law
From Faraday’s experiment, ∇ × E = `-\frac{dB}{dt}`
Ampere’s Circuital law applied to steady magnetic field gives
∇ × H = J
Taking Divergance on both sides
∇. (∇× H) = ∇.J
But ∇× H = 0 for static
∇.J = `-\frac{d\rho _{v}}{dt}` ( from equation of continuity)
Let us take an arbitrary ‚G‘ quantity for solving above unrealistic limitation of `\frac{d\rho _{v}}{dt}`=0
∴ ∇× H = J + G
Taking Divergence
∇. (∇× H) = ∇.J + ∇G
⇒ ∇.G = -∇.J = `\frac{d\rho _{v}}{dt}`
We know that ∇.D = `\rho_{v}` (Guass’s Law)
⇒ ∇.G = `\frac{d(∇.D)}{dt} = ∇. \frac{dD}{dt}`
We get G=`\frac{dD}{dt}`
Thus, Ampere’s law becomes
∇× H = J + `\frac{dD}{dt}`
It also agrees with continuity equation.
