Starting from Ampere’s circuital law, find the B field, total flux,and inductance of coaxial cable between the conductor and the shielding.

Sure, let's start with Ampere's circuital law. This law states that the integral of the magnetic field (B) around a closed loop is equal to the product of the current enclosed by the loop and the permeability of the medium. Mathematically, it is represented as:

∮ B . dl = μ0 * I

where:

• B is the magnetic field
• dl is the length element along the closed path
• μ0 is the permeability of free space
• I is the current enclosed by the path

Now, let's consider a coaxial cable with inner conductor radius 'a' and outer conductor (shielding) radius 'b'. The current I flows in the positive z direction in the inner conductor and returns in the negative z direction in the outer conductor.

1. Magnetic Field (B): The magnetic field at a distance 'r' from the axis inside the cable (a < r < b) due to the current in the inner conductor can be found using Ampere's law. Consider a circular path of radius 'r' and length 'dl' = rdφ. Applying Ampere's law:

   ∮ B . dl = B * 2πr = μ0 * I => B = (μ0 * I) / (2πr)

2. Total Flux (Φ): The total magnetic flux through the surface of the inner conductor is given by the integral of B over the surface area of the conductor. For a coaxial cable, this is the difference in flux between the outer and inner surfaces:

   Φ = ∫ B . dA = ∫ ((μ0 * I) / (2πr)) * 2πr dr from 'a' to 'b' => Φ = μ0 * I * ln(b/a)

3. Inductance (L): The inductance of the cable is defined as the ratio of the total flux to the current:

   L = Φ / I = μ0 * ln(b/a)

So, the magnetic field, total flux, and inductance of a coaxial cable can be found using Ampere's circuital law.