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 find the magnitude of the magnetic field at point P, we can use Ampere's law. Ampere's law states that the line integral of the magnetic field around a closed path is equal to the product of the current passing through the path and the permeability of the medium. In this case, we can consider a circular path of radius r around the cable's axis.
To apply Ampere's law, we need to determine the current passing through the path. Since the current in the center conductor is 2 Amps coming out of the page and the current in the outer conductor is 1 Amp going into the page, the net current passing through our circular path is 2 - 1 = 1 Amp.
We also need to know the permeability of the medium. In this case, since the cable is likely surrounded by air or vacuum, we can assume the permeability is equal to the permeability of free space, which is denoted by μ₀ and has a value of approximately 4π x 10^-7 Tesla meter per Ampere (T·m/A).
Now, we can use Ampere's law to calculate the magnetic field at point P.
The equation for Ampere's law can be written as:
∮ B · dl = μ₀ * I
where ∮ represents the line integral around the closed path, B is the magnetic field, dl is an infinitesimal element along the path, μ₀ is the permeability of free space, and I is the current enclosed by the path.
Since the magnetic field B and the infinitesimal element dl are parallel, their dot product B · dl simplifies to B * dl. Thus, the equation becomes:
B * ∮ dl = μ₀ * I
To evaluate the line integral, we replace ∮ dl with the circumference of our circular path, 2πr. Substituting this back into the equation, we have:
B * 2πr = μ₀ * I
Solving for B, we get:
B = (μ₀ * I) / (2πr)
Now, we can plug in the values we have:
μ₀ ≈ 4π x 10^-7 T·m/A
I = 1 Amp
r = 5 mm = 0.005 m
B = (4π x 10^-7 T·m/A * 1 A) / (2π * 0.005 m)
= (4π x 10^-7 * 1) / (2π * 0.005)
= (2 x 10^-7) / (0.01)
= 2 x 10^-5 Tesla (T)
Therefore, the magnitude of the magnetic field at point P, which is 5 mm from the cable's axis, is 2 x 10^-5 Tesla (T).