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Homework Help: How does this expansion work?

  1. Oct 27, 2006 #1
    OK, I'm given a magnetic field and told the field is cylindrically symmetric. Obviously, divergence of B must be zero, so taking the divergence in cylindrical coordinates,

    (1/r)(B_r) + (dB_r/dr) = -(dB_z/dz)

    where B_r is the radial component of the vector and B_z is the z component of the vector.

    Then, taking a Taylor expansion about r = 0, z = z_0,

    B_r = B_r(0, z_0) + r*(dB_r/dr) + (z - z_0)*(dB_r/dz)

    Substituting in the condition from the divergence,

    B_r = -r*(dB_z/dz) + z(dB_r/dz)

    So, for z = z_0,

    B_r = -r*(dB_z/dz)

    But, my book says that for small values of r, and z = z_0

    B_r = -(r/2)*(dB_z/dz)

    Anyone know where that factor of 1/2 might be coming from?
     
  2. jcsd
  3. Oct 28, 2006 #2

    OlderDan

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    Doesn't the zero divergence condition demand that B_r(0, z_0) = 0? And I don't see where you get this
    B_r = -r*(dB_z/dz) + z(dB_r/dz)
    What happened to the z_0?

    I see a 1/2 in there if you incorporate this.
     
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