Magnetic Field at a point outside a solenoid

[zorro]

Someone please explain me why is the magnetic field at a point outside a current carrying solenoid 0?
I read many books but can't understand what they mean.
I understood that the magnetic field at a point in between to adjacent coils is 0 as the 2 fields oppose each other. Why and how is it 0 outside the solenoid?

 

[Feldoh]

It's only zero to a good approximation. In actuality if you were to measure it you would see a small (relative to the field inside) magnetic field that loops around in a manner similar to a bar magnet.

 

[zorro]

Yes I know that it is just an approximation, but can't understand how that approximation is reached. Someone please help!

 

[Born2bwire]

Well, it has to do with the fact that for an infinite solenoid, you have sources of infinite size. Sources of infinite size generally indicate that their resulting fields are independent of one or more dimensions. For example, a point source charge gives rise to a field that falls off as 1/r^2. A line source of charge gives rise to a field of 1/\rho (where \rho is the perpendicular distance from the line source). A sheet of charge gives rise to a static field.

So what happens when we look at our solenoid? Well, one way to look at a solenoid is that it is composed of a set of infinitely tall strips of transverse current that are infinitesimally wide. That is, instead of thinking of the solenoid as an inifinite set of current loops, we divide up the loops into sections of d\phi that are infinitely long.

So now I have a current strip that has dimensions of -\infty < z < \infty by r*d\phi and the current runs in the \phi direction (where r is the radius of the solenoid). It turns out that the magnetic field contributed by this strip is constant,

$$d\mathbf{B} = \frac{\mu_0 K}{2\pi} d\phi \hat{z}$$

Now for a cylindrical solenoid, any cross-sectional view of the solenoid is one where we will see two of these strips but with currents running in opposite directions. On the interior, these currents work together but on the exterior they oppose. Since they are independent of distance (since they are infinite sources) then the two current strips perfectly cancel each other out outside of the solenoid.

Actually, I found a link to a paper that discusses part of what I stated above: http://www.physics.princeton.edu/~mcdonald/examples/EM/espinoza_ajp_71_953_03.pdf

To make sense of what I said above in relation to Figure 1, let's assume the observation point P is the center of the solenoid. Then the currents that we see at P at any point along the solenoid all run in the same direction with respect to the direction \phi. However, if P is outside the solenoid, then in a cross-sectional view, the nearest current element may run parallel to \phi and the farther one will run opposite. If I had a tablet PC it would be easy for me to draw some pictures to make this clear but give it a think and it should become apparent.

 

[zorro]

Thanks born2bwire!