Deriving finite solenoid field by integrating thin shell equation

[demon_samuel]

Dear all,

i met a problem that i couldn't get the on axis field of a finite solenoid that integrating from a thin shell solenoid equation.

The thin shell equation link:
http://www.netdenizen.com/emagnet/solenoids/thinsolenoid.htm

And the on axis finite solenoid equation link:
http://www.netdenizen.com/emagnet/solenoids/solenoidonaxis.htm

If i was right, the integral of (a/sqrt (a*a+r*r)) dr should be a*ln(r+sqrt(r*r+a*a)), if so i couldn't figure out how to get the (r2-r1) term in the second link equation.

Thank you for your attention

Samuel

 

[kuruman]

To get the second expression from the first, you need to integrate over dr from r₁ to r₂. You do this by adding "solenoids" of radius r, thickness dr, and length l carrying current di. According to the first expression, the contribution from one such solenoid is

$$dB = \frac{\mu_{0}N di}{2 l}[\frac{x_{2}}{\sqrt{x^{2}_{2}+r^{2}}}-\frac{x_{1}}{\sqrt{x^{2}_{1}+r^{2}}}]$$

Before you integrate, you need to express di in terms of dr. Hint: Think in terms of current density.

 

[demon_samuel]

Thank you very much, i got the answer.