Magnetic field expansion in cylindrical coordinate

[demon_samuel]

Homework Statement

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.

Homework Equations

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.

 

[Jhero]

Thank you for sharing your work on the Taylor expansion for magnetic field in cylindrical coordinates. It seems like you have a good understanding of the mathematical relationships involved. To answer your question about the "-1/4 B'' r^2" term, let's take a closer look at the equations you have listed.

Equation (1) represents the divergence of the magnetic field, which must be zero in a cylindrical coordinate system. Equation (2) represents the relationship between the z and theta components of the magnetic field, and equation (3) represents the relationship between the r and z components. Finally, equation (4) represents the relationship between the r and theta components.

Now, let's consider the Taylor expansion for the magnetic field in just the z direction. This would be equivalent to setting all other components (r and theta) to zero. In this case, equations (1), (3), and (4) would all be satisfied, but equation (2) would not. This is because the z component of the magnetic field is no longer equal to the theta component in this case.

To account for this discrepancy, we introduce the "-1/4 B'' r^2" term in the Taylor expansion. This term represents the difference between the z component and the theta component of the magnetic field, and it is necessary to satisfy equation (2) and maintain the overall consistency of the magnetic field equations.

I hope this helps to clarify why the "-1/4 B'' r^2" term is included in the Taylor expansion for the magnetic field in just the z direction. Keep up the good work on your research!