How Can a Toroid Approximate a Solenoid in Limiting Conditions?

[Chris W]

Hi everyone.
I need help please.
I am working on problems with solenoids and Toroids
I have solution for the solenoid:
B = μo i n

And toroid:
B = (μ o i n)/ (2Π r)
Also, I know that the magnetic field is the function of r namely: B = B(r)

r- radius of the Ampere’s path
n – number of loops per unit length
i-Current
μo – constant

My problem is:
Using the solution for the toroid, show that for the large toroid the answer can be approximated as the solenoid on the very small piece of the toroid.
I know that I have to play with limits. Something like:
a - inner radius of the toroid,
b – outer radius of the toroid,
∆a - the difference between radius a and radius b.
I think I have to take a limit when ∆a goes to 0 and in this way radius a will approach radius b. in this way the solution for the toroid SHOULD be the solution for the solenoid (on the small length L of course)
I don’t know how to set it up. How to get from the toroid solution to the solenoid solution using limits or (other technique)

Thanks for help
Chris W

 

[Chris W]

Can anyone please help me here... thanks
Chris W

 

[alphysicist]

Hi Chris W,

Your toroid magnetic field equation is not quite right. It should be:

$$B = \frac{\mu_0 i N }{2\pi r}$$

where N is the total number of turns (not turns per length). Notice that $N/(2 \pi r)$ is in a way similar to the n in the solenoid formula; but what is the difference? If you then think about your limiting process that should help you get the result.

 

[physixguru]

http://en.wikipedia.org/wiki/Solenoid
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/toroid.html

 

[Chris W]

Thanks Guys. I love this forum
!
Chris W