Finding B-Field Of Solenoid Slightly Off Axis?

[jasonpatel]

If you have a solenoid positioned along the z axis...

...how would one find the b-field at slight deviations of x and y?

I have been googling for hours and can't find anything other than the fact that it is very difficult!

Thanks guys!

 

[bp_psy]

This is kind of a hard problem that involves elliptic integrals.
Here is paper that shows you how to do it .

http://ntrs.nasa.gov/search.jsp?R=19980227402

Also this might help.
http://www.netdenizen.com/emagnettest/offaxis/?offaxisloop

 

[jasonpatel]

Interesting read, but extremely complicated! Haha, i was wondering if there was a way to taylor expand around the z axis with infinitely small deviations from the z axis (i.e. x+ε and y+ε)

Any thoughts?

 

[bp_psy]

Interesting read, but extremely complicated! Haha, i was wondering if there was a way to taylor expand around the z axis with infinitely small deviations from the z axis (i.e. x+ε and y+ε)

Any thoughts?

What exactly do you want to expand?
Note that the magnetic field inside a sufficiently long solenoid is in fact homogeneous and there is no need to expand anything, Also the need for complicated math and numerical integration techniques arises even on axis for a short finite solenoid since the fields at the ends diverge.

http://en.wikipedia.org/wiki/Soleno...ctor_potential_for_finite_continuous_solenoid

 

[jasonpatel]

Well, let me get a little more specific then.

My actual problem was to find the b-field of a helix coil along the axis of symmetry (z-axis), and I did. The B_x, B_y and B_z components are all a function of z and no other variables.

Then I was asked to find the b-field at very small deviations from the z-axis, x+ε and y+ε. I was also given the advice to do some "Taylor expansion of the field".

But i don't have a clue of how to do that! Any help??

 

[jasonpatel]

Just to make things very clear I have attached a pdf of the integrals I will be computing. So, I am wondering if I should expand ∅ with a taylor expansion because of the assumption that displacements of ∅ will be very small i.e. (∅ + ε) and (∅ - ε)

I am hoping this expansion of ∅ will create a simple integral and the answer of which will give the b-field in terms of (x,y,z) with the assumption that displacements in the (x,y,z) will be very small (x+ε,y+ε,z+ε).

Does this sound reasonable or am I totally off? Also, how many terms in the taylor expansion should I keep?