Magnetic field expansion in cylindrical coordinate

1. Oct 8, 2009

demon_samuel

1. The problem statement, all variables and given/known data
Dear all, i am figuring the taylor expansion for magnetic field in cylindrical coordinate (r,theta,z). The taylor expansion is simple, however if i want to express the magnetic field in just z direction, i don't know to start. It is because i seen some books written:

Bz = Bo + B' z + 1/2B'' z^2 - 1/4 B'' r^2 +.... , where B' = partial Bz respect to z
B'' = second partial Bz respect to z

I couldn't figure out why the '-1/4 B'' r^2' term is here.

2. Relevant equations

3. The attempt at a solution
Sorry that i could not how to start, so i just can show my work on the the taylor series of Bz = Bo + B' z + B''z^2/2! +B''' z^3/3! +....
And i know that if the magnetic field is cylindrical , the curl and div of B is zero, and i get the relationship that

1/r partial (rB_r) respect to r + 1/r partial (B_theta) respect to theta + partial (B_z) respect to z = 0 --- (1) (from div B=0) AND
1/r partial (B_z) respect to theta = partial (B_theta) respect to z --- (2) (from curl B= 0) AND
partial (B_r) respect to z = partial (B_z) respect to r --- (3) (from curl B= 0) AND
partial (rB_theta) respect to r = partial (B_r) respect to theta --- (4) (from curl B= 0)

Can anyone help me and give me some ideas? Thank you for your help and attention.

Last edited: Oct 8, 2009