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Magnetic field in an offset hole in a conductor cylinder

  1. Dec 12, 2016 #1
    1. The problem statement, all variables and given/known data
    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.
    2. Relevant equations
    ##\oint B \cdot dl = \mu_0 I_\text{enc}##

    3. 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.

    Attached Files:

  2. jcsd
  3. Dec 13, 2016 #2


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    Science Advisor
    Gold Member

    I'm not sure what s is. Your expressions for B should be involve the radii a and b. As for direction, write B as a vector instead, starting with Ampere's law written properly as $$\oint{\vec B \cdot \vec{dl}}=\mu_0 I_{enc}.$$Think about what contour you use in the integral, and what direction ##\vec B## takes along that contour.
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