Need a Review of Solenoid Equations and their Derivation

[Prince Rilian]

I need a review of solenoid equations, such as equations for bar solenoids, toroidal solenoids, and toroidal solenoids with an air gap within them. I think that somewhere along the line I may have learned something incorrect, so I would like to kill any douts that I have by ascertaining myself of the correct equations.

 

[BruceW]

the bar (straight) solenoid is the simplest. You can derive the magnetic field by using Maxwell's equations (in integral form). For the toroidal solenoid, I guess it is more complicated, unless the major radius is much greater than the minor radius, in which case I guess it would be approximately the same as in the case of the straight solenoid.

 

[Prince Rilian]

I know that the equation for the inductance of a bar-shaped air-core solenoid is

L = μ₀AN²/l,​

assuming that both of the dimensions of the area component are much less than the variable "l". For a toroidal solenoid, I know that you can use the same form of equation if you use the circumference of the torus in place of "l".

The problem in my understanding seems to come about once a magnetic core is inserted into the solenoid...