How can we prove that the magnetic field will to the axis of the coil?

Sep 2019
It’s a well known fact that For a very long coil the field exists practically only in the interior. There it is uniform and parallel to the axis of the coil.

Why? How? Almost, every book (really every book) takes that fact as granted and then they solve for the magnitude of magnetic field by drawing a rectangular Amperian Loop.

David J. Griffiths does the prove but I find his proof very very personal. Here are the images, have a look:

Solenoid 1
Solenoid 2
Solenoid 3
Solenoid 4

So, is there any other proof of why the magnetic will be uniform and parallel to the axis of the solenoid?
Last edited:
Jun 2016
David J. Griffiths proof makes use of several arguments which it is assumed the reader is familiar with.
To a certain extent this is inevitable, otherwise the proof could end up taking up a book, rather than just a few pages.
But this does then require someone who arrives at his proof without the assumed knowledge to seek additional explanation.
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Sep 2019
I have found another proof of the solenoidal field in the book Electromagnetic Fields and Waves by Paul Lorrain. The proof goes like this:



Please help me in understanding the first point, that is how by symmetry $\mathbf B$ will be independent of $z$ and $\phi$ ? Symmetry is always a problem for me, I don’t understand when any book simply writes “by symmetry...” Please help me in seeing in it.