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,(adsbygoogle = window.adsbygoogle || []).push({});

(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?

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# How does this expansion work?

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