1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
    upload_2016-12-12_17-23-10.png
    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

    marcusl

    User Avatar
    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted