Provide the derivation of Ampere-Maxwell Law.

The Ampere-Maxwell Law is an equation in electromagnetism that describes the relationship between the electric current and the magnetic field. It is given as:

where ∇ × B is the curl of the magnetic field B, J is the electric current density, ε0 is the permittivity of free space, and μ0 is the permeability of free space.

The derivation of the Ampere-Maxwell Law involves using the equations of Maxwell's equations, which are a set of four fundamental equations that describe the behavior of electric and magnetic fields. The equations are:

1. Gauss's Law for Electric Fields: ∇ · E = ρ/ε0
2. Gauss's Law for Magnetic Fields: ∇ · B = 0
3. Faraday's Law: ∇ × E = - ∂B/∂t
4. Ampere's Law: ∇ × B = μ0 J

To derive the Ampere-Maxwell Law, we begin with Ampere's Law and apply Faraday's Law to it:

Using the vector identity ∇ × (∇ × B) = ∇(∇ · B) - ∇²B, we get:

∇(∇ · B) - ∇²B = μ0 ∇ × J

Since ∇ · B = 0 (from Gauss's Law for Magnetic Fields), the first term on the left-hand side vanishes, leaving:

∇²B = - μ0 ∇ × J

Next, we use Faraday's Law to substitute for ∇ × E in the expression for J:

∇ × J = ∇ × (ε0 ∂E/∂t)

Substituting this into the above equation, we get:

∇²B = - μ0 ε0 ∂²B/∂t²

This is the wave equation for the magnetic field, which describes how the field propagates through space at the speed of light. By taking the curl of Faraday's Law, we can similarly derive the wave equation for the electric field.

Finally, we can combine the wave equations for the electric and magnetic fields to obtain the full set of Maxwell's equations:

∇²B = μ0 ε0 ∂²B/∂t² ∇²E = μ0 ε0 ∂²E/∂t²

Ampere-Maxwell Law

These equations describe how electric and magnetic fields propagate through space and interact with each other. The Ampere-Maxwell Law is a specific case of these equations, where the current density is constant and there are no charges in the region of interest.

Also check: