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.

Homework Statement

A coaxial cable consists of an inner solid conductor of radius $r_1$, and and outer conducting cylindrical shell of inner radius $r_2$ and outer radius $r_3.$ In both conductors, the current equals $I_0$, 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 $j_1=C_1 r$ and in the outer cable as $j_2=C_2 r$.

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

a) $R < r_1$
b) $r_1 < R < r_2$
c) $r_2 < R < r_3$
d) $R > r_3$

Homework Equations

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

The Attempt at a Solution

a) We consider a circle of radius $R < r_1$ 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 $$\oint \bar{B} \cdot d\bar{s} = 2\pi B R.$$
For the RHS we need to find the total current through the circle.
$$\begin{align*} I &= \iint\limits_{circle} j_1 dA \\ &= \int_0^{2\pi} d\theta \int_0^R C_1 r^2 dr \\ &= \frac{2}{3} \pi C_1 R^3 \end{align*}$$

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

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



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

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

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

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

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



c) The total current is zero, therefore, so is the magnetic field.
$$B = 0$$

 

[kuruman]

Everything looks OK to me.