Zero B Field Inside within Hollow Wire

[Amrator]

Homework Statement

A long, thin wire carrying constant current I1 = 2 A into the page is surrounded by a concentric cylindrical hollow wire of inner radius a = 0.12 m, and outer radius b= 0.26 m, carrying total current I2 = 4 A directed out of the page, as shown. Assume the current in the cylindrical hollow wire is distributed uniformly over its cross-sectional area.

At what radius is B = 0 in the region a < r < b inside the hollow wire?

(A) .19 m
(B) .20 m
(C) .21 m
(D) .25 m
(E) The magnetic field is not zero anywhere inside the hollow wire.

Homework Equations

Ampere's Law

The Attempt at a Solution

Using ratios:
$I_r → 2π(r-a)$
$I_b → 2π(b-a)$
$I_{enc} = I_2 \frac{I_r}{I_b} - I_1 = (4A) \frac{r-a}{b-a} - I_1$
$b-a = .14$
$\frac{r-.12}{.14} = 1/2$
$r = .19 m$

The answer however is not A. What am I doing wrong?

 

[MisterX]

$$|\mathbf{B}| = \frac{\mu_0 I_{enc}}{2\pi r} $$

I am not sure you are correctly expressing $I_{enc}$ as a function of $r$. It is going to be related to the area of an annulus. So I would expect to see $r^2$ somewhere in your work.

 

[rude man]

The answer however is not A. What am I doing wrong?

Apply Ampere's law anywhere in the region a < r < b. What do you get?