Taylor Expansion of A Magnetic Field

[jasonpatel]

Quick question about Taylor expansions that I am getting pretty confused about. Let's say using biot savart I want to find the field of a INFINITE helix (http://en.wikipedia.org/wiki/Helix) along the axis but with very slight displacements of x and y (x+ε,y+ε). Here is a series of steps I will go through:

1. set up biot savart integral along z-axis (x=y=0)

2. compute biot savart integral with respect to theta

3. evaluate integral from -∞ to +∞ because infinite helix

4. obtain a function of Bx(z), By(z) and Bz(z)

So, when should I do the Taylor expansion to find {Bx,By,Bz} at very slight x and y? Can I even do this with a function that doesn't have x or y in it?

Do I need to revaluate this biot savart integral with x,y≠0? to obtain an expression of B like{Bx(x,y,z),By(x,y,z),Bz(x,y,z)}? If so when do I taylor expand?

Thanks in advance

 

[Jano L.]

If I understood you correctly, you want to find the field on line x = ε, y =ε just from the function $\mathbf B(0,0,z)$ defined on the line x=0,y=0,z=0. Is that right?

You can write the Taylor theorem in this way:

$$B_z(ε,ε,z) = B_z(0,0,z) + \frac{\partial B_z}{\partial x} ε + \frac{\partial B_z}{\partial y} ε$$

But the problem is how to find out the derivatives. I do not think you can find them just from $\mathbf B(0,0,z)$.

It seems you will have to use Biot-Savart for line ε,ε,z directly. However, you may end up with nasty integral. And perhaps here you can simplify them by expanding the function under integral up to first or second order terms in ε.

 

[Jano L.]

Heureka, in fact one can use small trick to simplify it. The line is in free space and the field is static, so the curl of B is zero. This implies that

$$\frac{\partial B_z}{\partial x} = \frac{\partial B_x}{\partial z}$$

and

$$\frac{\partial B_z}{\partial y} = \frac{\partial B_y}{\partial z}$$

so you can write

$$B_z(ε,ε,z) = B_z(0,0,z) + \frac{\partial B_x}{\partial z}ε + \frac{\partial B_y}{\partial z}ε$$

and this can be calculated just from $\mathbf B(0,0,z)$

Perhaps similar trick using $\nabla\cdot\mathbf B = 0$ and helical symmetry may help to find out the components $B_x, B_y$.

 

[jasonpatel]

Hi thanks for your reply. I do think I understand what you are saying but just to make sure I attached a pdf of what I plan to do (only the x component is shown in the pdf...ill be doing the y,z components as well).

Basically ill perform the biot savart integral with x, y and z as passive variables and integrate with respect to dθ.

ALso, where did you get that formulation of the taylor series? I have never seen it before. Thanks again for your help!

 

[jasonpatel]

Also I am not only looking at the z component of the B field, I am also looking at the x and y component as well.

Actually the field of B[0,0,z] is not static, i have done the calculation already:

1. B-z component is static along the z axis

2. B-y component is sinusoidal along the z axis

3. the B-x component is sinusoidal along the z axis

This is all done along the z axis so x=y=0

 

[jasonpatel]

Any ideas? I am kind of stuck here esp with the integrals in full {x,y,z} space (not just on the axis x=y=0)