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Homework Statement
A long (infinite) wire (cylindrical conductor of radius R, whose axis coincides with the z axis carries a uniformly distributed current I in the +z direction. A cylindrical hole is drilled out of the conductor,
parallel to the z axis, (see figure above for geometry). The center of the hole is at x = b , and the radius of the hole is a. Determine the magnetic field in the hole region.
Homework Equations
##\oint B \cdot dl = \mu_0 I_\text{enc}##
The Attempt at a Solution
I'm pretty sure that I have the magnitude of the field, but I'm unsure how to get the direction. For the magnitude, I took a superposition of a cylinder with current density ##J## and a cylinder of current density ##-J## where the hole is. The magnitude of the magnetic field of each cylinder is $$B_1=\frac{\mu_0 J}{2}s_1 $$ and $$B_2=- \frac{\mu_0 J}{2}s_2 $$ respectively. Adding magnitudes together gives $$B_1+B_2=\frac{\mu_0 J}{2}(s_1-s_2)$$ and since ##s_1-s_2 = b## and ##J=\frac{I}{2 \pi (R^2-a^2)}## for the magnitude in the hole I get $$B_1+B_2=\frac{\mu_0 I b}{2 \pi (R^2-a^2)}$$
My professor said it would be easier to find the direction if I convert to cartesian coordinates, but my book says that the ##\hat{s}## direction in cartesian coordinates is ##cos\phi \hat{x} +sin \phi \hat{y}## but that would seem to give $$B_1+B_2=\frac{\mu_0 I b}{2 \pi (R^2-a^2)} (cos\phi \hat{x} +sin \phi \hat{y})$$ which definitely does not make sense.