Answer to Question #89985 in Electricity and Magnetism for Aima sajid

Question

There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.” -Hamlet

A long solenoid of length L and radius R has a total of N turns and carries a current I. The solenoid is held firmly in place and cannot rotate.
a. Use Ampere’s law to determine the magnetic field inside the solenoid (far from itsends).

Answer 1

Consider a solenoid of length L.

According to the law of Bio-Savart-Laplace:

$$dB=\frac{\mu_0}{4\pi}\frac{I\,drsin{\alpha}}{r^2}$$

Then the field of one turn:

$$B=\frac{\mu_0}{4\pi}\frac{I*4\pi}{2}\int_{\alpha_1}^{\alpha_2} d\alpha\, sin{\alpha}=\mu_0\frac{I}{2}(cos(\alpha_1)-cos(\alpha_2))$$

In the solenoid N turns, therefore, the total field is equal to:

$$B=\mu_0N\frac{I}{2}(cos(\alpha_1)-cos(\alpha_2))$$