Question

Consider a coaxial cable (like the kind that is used to carry a signal to your TV). In this cable, a current I runs in one direction along the inner central wire, and the same current I runs in the opposite direction in the hollow conductor. How does the magnetic field B outside the entire cable (outside both the inner wire as well as the hollow conductor) vary as a function of distance away from the cable

Answer 1

Answer:

0 < r < r_exterior     B_total = $\frac{\mu_o I}{2\pi r}$

r > r_exterior            B_total = 0

Explanation:

The magnetic field created by the wire can be found using Ampere's law

        ∫ B. ds = μ₀ I

bold indicates vectors and the current is inside the selected path

           

outside the inner cable

          B₁ (2π r) = μ₀ I

          B₁ = $\frac{\mu_o I}{2\pi r}$

the direction of this field is found by placing the thumb in the direction of the current and the other fingers closed the direction of the magnetic field which is circular in this case.

For the outer shell

for the case   r> r_exterior

         

           B₂ = \frac{\mu_o I}{2\pi  r}

This current is in the opposite direction to the current in wire 1, so the magnetic field has a rotation in the opposite direction

for the case r <r_exterior

in this case all the current is outside the point of interest, consequently not as there is no internal current, the field produced is zero

           B₂ = 0

Now we can find the field created by each part

0 < r < r_exterior

          B_total = B₁

          B_total = $\frac{\mu_o I}{2\pi r}$  

r > r_exterior

          B_total = B₁ -B₂

          B_total = 0