Question

A long and an ideal solenoid of length l, radius R and number of turns N carries a steady current I. Determine the flux linked with a circular surface C of radius r in each of the following cases: 

 

Fig i : r > R , Plane of coil ⊥ axis of the solenoid 

 

Fig ii : r < R , Plane of coil ⊥ axis of the solenoid 

 

Fig iii : r >R , Normal to the plane of coil subtends an angle 30^o with the axis of the solenoid

Answer 1

Fig i : 

Here θ = 0, 

\(\phi_{in} = B_{in}A\cos\theta\) 

\(\phi_{in} = \frac{\mu_0NI}{l}\pi R^ 2\cos0\) 

\(\phi_{in} = \frac{\mu_0NI}{l}\pi R^2\)

\(\Phi_{out} = 0\) (Magnetic field outside the solenoid is zero.)

\(\phi_c = \phi_{in} + \phi_{out}\) 

\(\phi_c = \frac{\mu_0NI}{l}\pi R^2\)

Fig ii : 

\(\phi_{in} = B_{in}A\cos\theta\) 

\(\phi_{in} = \frac{\mu_0NI}{l}\pi r^2\cos0\)  

\(\phi_{in} = \frac{\mu_0NI}{l}\pi r^2\) 

Fig iii : 

\(\phi_{in} = B_{in}A\cos\theta\) 

\(\phi_{in} = \frac{\mu_0 NI}{l}\pi R^2\cos30 \) 

\(\phi_{in} = \frac{\sqrt{3}\mu _0NI}{2l}\pi R^2\)