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?

Trivial problem, come on. The E effective for ammpers' law is 1am out-2amps in= -1 amp (inward)

Use ampere's law.

For a more difficult version, see this:
http://physics.bu.edu/~duffy/semester2/d14_ampere_cylinder.html

To find the magnitude of the magnetic field at point P, we can use Ampere's Law. Ampere's Law states that the integral of the magnetic field around a closed loop is equal to the product of the current passing through that loop and the permeability of the medium.

In this case, we need to consider a circular loop centered at the axis of the coaxial cable, passing through point P and having a radius of 5mm. Let's call this loop L.

The current passing through the coaxial cable is the combination of the currents in the center conductor and the outer conductor. However, the direction and distance of each current from the loop L will affect the magnetic field at point P.

Since the center conductor current of 2A is coming out of the page, it will create a magnetic field that circulates counterclockwise around it. On the other hand, the outer conductor current of 1A going into the page will create a magnetic field that circulates clockwise around it.

To find the contribution of each conductor to the magnetic field at point P, we can apply the Biot-Savart Law, which relates the magnetic field at a point to the current element and its distance. We need to calculate the magnetic fields created by the center conductor and the outer conductor separately and then add them together vectorially to obtain the total magnetic field at point P.

Let's calculate the two contributions individually:

1. Magnetic field due to the center conductor:
We can apply the Biot-Savart Law to find the magnetic field at point P due to the current in the center conductor. The formula is given as:

dB = (μ₀/4π) * (Idl × r̂) / r²

where:
- μ₀ is the permeability of free space (4π × 10⁻⁷ T·m/A)
- Idl is the current element
- r̂ is the unit vector pointing from the current element to the point P
- r is the distance between the current element and point P

The current element Idl is equal to the total current in the center conductor, which is 2A. The distance r is 5mm since point P is 5mm away from the axis of the cable.

Since the magnetic field at point P is the vector sum of the magnetic fields produced by all the current elements along the center conductor, we need to integrate the above expression over the length of the center conductor.

2. Magnetic field due to the outer conductor:
Similarly, we can apply the Biot-Savart Law to find the magnetic field at point P due to the current in the outer conductor. The formula remains the same, except the current element Idl is equal to the total current in the outer conductor, which is 1A.

Again, we need to integrate this expression over the length of the outer conductor to find the total magnetic field at point P.

Finally, we can add the magnetic fields due to the center conductor and the outer conductor vectorially to obtain the total magnetic field at point P.

It is worth noting that these calculations assume the coaxial cable is infinitely long, and the fields decay as you move away from the cable. Also, the radius values mentioned in the question are helpful for determining the currents and the geometry of the cable but are not directly used in calculating the magnetic field at point P.