How is the Magnetic Field Expression Derived in This Coil Configuration?

[nabliat]

http://i29.tinypic.com/2rdiuye.jpg

i can't understand how they got the first expression

in a normal coil the distance is r
but here they have a root in the denominator
like they used Pythagorean theorem and multiplying by sinus

and i can't understand where they do it here in order to get the
field expression
??

 

[fantispug]

The first expression is the magnetic field due to a current loop of current NI, radius R1 at a point on the axis of symmetry a distance x from the centre. (That is, because the solenoid is so far away we treat it just as a current loop). You can calculate this directly from the Biot-Savart law.

I can't quite follow what you're saying, but you are right in that the square root comes from a sine; by (cylindrical) symmetry the only component of the field that isn't canceled acts along the axis of symmetry. So we just take this component to get the sine.

(Have a look in your textbook for the magnetic field due to a current loop for the details).

 

[nabliat]

i have this formula for a magnetic inside a coil

$$B=\frac{\mu _0NI}{l}$$

this is a cross section of the system.
to have a mutual induction
i would need to have some current in one of the wires

i don't have it here

??