Consider a coaxial cable with 2Amps in the center conductor coming out of the page, and 1Amp in the outer conductor going into the page. The center conductor has a radius of 1mm, the outer conductor's inner radius is 2mm, and the outer conductor's outter radius is 3mm. What is the magnitude of the magnetic field at point P which is 5mm from the cable's axis?

To determine the magnitude of the magnetic field at point P, you can employ Ampere's law. Ampere's law states that the line integral of the magnetic field around a closed curve is equal to the product of the permeability of free space (μ₀) and the total current passing through the surface that the closed curve encloses.

Here's how you can apply Ampere's law to solve this problem:

1. Identify a closed curve that passes through point P and encloses the currents in the cable. In this case, you can consider a circular path centered on the coaxial cable axis with a radius of 5 mm.

2. Calculate the total current passing through the surface enclosed by the circular path. The total current is the sum of the currents in the center conductor and the outer conductor. In this case, it would be 2 Amps (center conductor) + 1 Amp (outer conductor) = 3 Amps.

3. Compute the line integral of the magnetic field around the closed curve. Since the current is constant along the coaxial cable, the magnetic field will vary with the distance from the cable's axis. You can consider different cases based on the location of the circular path:

- Inside the center conductor (r < 1 mm):
The magnetic field is given by the formula: B = (μ₀ * I) / (2π * r)
Substitute the values: B = (4π * 10^(-7) T⋅m/A * 2 A) / (2π * 0.005 m) = 0.4 T

- Between the center and outer conductors (1 mm < r < 2 mm):
There is no current passing through this region, so the magnetic field is zero.

- Inside the outer conductor (2 mm < r < 3 mm):
Similar to the case inside the center conductor, the magnetic field is given by B = (μ₀ * I) / (2π * r), where I is the current passing through this region (1 Amp).
Substitute the values: B = (4π * 10^(-7) T⋅m/A * 1 A) / (2π * 0.005 m) = 0.2 T

- Outside the outer conductor (r > 3 mm):
There is no current passing through this region, so the magnetic field is zero.

4. Add up the magnetic field contributions from each region. In this case, you only have non-zero contributions from the inside of the center conductor and the inside of the outer conductor.
B_total = B_center conductor + B_outer conductor = 0.4 T + 0.2 T = 0.6 T

Therefore, the magnitude of the magnetic field at point P, which is 5 mm from the cable's axis, is 0.6 Tesla.