Magnetic field in coaxial cables

A_B

I think I solved this problem correctly, I just want to check because it's an interesting problem, and it's from a past exam.


1. The problem statement, all variables and given/known data
A coaxial cable consists of an inner solid conductor of radius [itex]r_1[/itex], and and outer conducting cylindrical shell of inner radius [itex]r_2[/itex] and outer radius [itex]r_3.[/itex] In both conductors, the current equals [itex]I_0[/itex], but they are in opposite directions. Also, the current distribution is not uniform, it increases linearly with distance from the central axis. In the inner conductor is varies as [itex]j_1=C_1 r[/itex] and in the outer cable as [itex]j_2=C_2 r[/itex].

Calculate the magnetic induction (which I take means magnetic field strength) as function of [itex]R[/itex], the distance from the central axis, and [itex]I_0[/itex] for:

a) [itex]R < r_1[/itex]
b) [itex]r_1 < R < r_2[/itex]
c) [itex]r_2 < R < r_3[/itex]
d) [itex]R > r_3[/itex]


2. Relevant equations
I think the static Ampère's law is the correct equation
[tex]\oint \bar{B} \cdot d\bar{s} = \mu_0 I[/tex]
Where I is the current through any surface which has the curve of the LHS as boundary.


3. The attempt at a solution

a) We consider a circle of radius [itex]R < r_1[/itex] centered at the central axis of the coaxial cable in the plane perpendicular to the current. The LHS of Amp's law for this circle gives [tex]\oint \bar{B} \cdot d\bar{s} = 2\pi B R.[/tex]
For the RHS we need to find the total current through the circle.
[tex]
\begin{align*}
I &= \iint\limits_{circle} j_1 dA
\\ &= \int_0^{2\pi} d\theta \int_0^R C_1 r^2 dr
\\ &= \frac{2}{3} \pi C_1 R^3
\end{align*}
[/tex]

If we take this integral from [itex]0[/itex] to [itex]r_1[/itex] , we know that the current through the circle is [itex]I_0[/itex]. Thus, we can figure out the constant [itex]C_1.[/itex]
[tex] C_1 = \frac{3I_0}{2\pi r_1^3} [/tex]
We put this in the previous formula for the current to obtain:
[tex] I = \left(\frac{R}{r_1}\right)^3 I_0 [/tex]

We plug it all into Ampère's law to obtain
[tex]B=\frac{\mu_0 R^2}{2\pi r_1^3}I_0[/tex]



b) When [itex]r_1 < R < r_2[/itex] the current through the circle is always [itex]I_0[/itex] and the countour integral still equals [itex]2\pi BR[/itex] so the magnetic field strength is
[tex] B = \frac{\mu_0 I_0}{2\pi R}[/tex]

c)The current through the circle is [itex]I_0[/itex] minus the current through the bit of the outer conductor that goes through the circle. We shall first calculate the current through this bit.
[tex]
\begin{align*}
I_2 &= \iint\limits_{r_2 \rightarrow R} j_2 dA
\\ &= \int_0^{2\pi} d\theta \int_{r_2}^{R} C_2 r^2 dr
\\ &= \frac{2}{3} \pi C_2 \left(R^2 - r_2^3\right)
\end{align*}
[/tex]

Taking this integral to [itex]r_3[/itex] should give [itex]I_0[/itex] so we can figure out the constant [itex]C_2[/itex].
[tex]C_2 = \frac{3I_0}{2\pi \left(r_3^3-r_2^3\right)}[/tex]

Putting it all together gives a total current through the circle of:
[tex]I = I_0 - I_2 = I_0 - \frac{\left(R^3 - r_2^3\right) I_0}{\left(r_3^3 - r_2^3\right)}[/tex]

Applying Amp's law gives
[tex] B = \frac{\mu_0 I_0}{2 \pi R} \left( 1- \frac{R^3 - r_2^3}{r_3^3 - r_2^3} \right)[/tex]



c) The total current is zero, therefore, so is the magnetic field.
[tex] B = 0[/tex]