Magnetic Field Outside a Finite Solenoid

mo09890

I'm trying to write a program to simulate the fields generated by a solenoid but I've hit a bit of a brick wall. There is a vast amount of examples and information on the field generated inside a solenoid however they all assume that the field outside is negligible and as such I have been having trouble finding any equations or information on the field outside of a solenoid.

Any pointers in the right direction would be greatly appreciated.

Bob S

The field on the axis of a solenoid is given by B = 0.5 μ₀ NI (cos β2 – cos β1) where β1 and β2 are the angles of the end of the coils subtended at a point on the solenoid axis. See Smythe “Static and Dynamic Electricity”, 3rd Edition, page 297.

So the field in the center of an infinite solenoid is B_center = μ₀ NI. The field inside the end of an half-infinite solenoid is B_end = 0.5 μ₀ NI

Because the line integral around the coil windings is (Ampere’s Law)

∫B dl = μ₀ NI

The field outside the solenoid has to be

B_outside = μ₀ NI - 0.5 μ₀ NI (cos β2 – cos β1)

Bob S

mo09890

Thanks for the very speedy reply Bob! I have one question though - is that the equation for the field outside the solenoid but still on the axis or will this work for any point in a 3D space? (Am i asking too much for it to be as simple as this for 3D? :P)

Bob S

Hi mo09890-
The result quantitativly applies to any line integral outside the coil. To get the actual B field anywhere outside the coil, you will have to choose the appropriate path of the line integral, meaning that although the line integral is correct, the value of B anywhere is not determined, except very close to the coil.
Bob S