Magnet Field of a Solenoid: Understanding the Derivation Step by Step

[zezima1]

Homework Statement

Attached file shows the problem statement.

The Attempt at a Solution

I can do this problem easily. Problem is that there's just one step in my book's derivation that I don't understand. It sets ρ equal to the hypotenuse in the triangle and then says that:

dl = ρ/sinθ dθ (1)

For some reason I can't get this result. If we look at the cosine of θ then:
cosθ = l/ρ
=>
l = ρcosθ
=>
dl/dθ = -ρsinθ
=>dl = -ρsinθdθ

Incosistent with (1). What am I doing wrong?

 

[tiny-tim]

hi zezima1!

… It sets ρ equal to the hypotenuse in the triangle and then says that:

dl = ρ/sinθ dθ (1)

For some reason I can't get this result. If we look at the cosine of θ then:
cosθ = l/ρ
=>
l = ρcosθ
=>
dl/dθ = -ρsinθ
=>dl = -ρsinθdθ

ah, but ρ isn't constant, h is …

differentiate cotθ = l/h ​