(a) Using Ampere’s circuital law, obtain the expression for the magnetic field due to a long solenoid at a point inside the solenoid on its axis.

(b) In what respect is a toroid different from a solenoid? Draw and compare the pattern of the magnetic field lines in the two cases.

(c) How is the magnetic field inside a given solenoid made strong?

a) Consider a symmetrical long solenoid having number of turns per unit length equal to n.

Let I be the current flowing in the solenoid, then by right hand rule, the magnetic field is parallel to the axis of the solenoid.
 

Field inside the solenoid:
Consider a closed path abcd.

Now, using Ampere’s circuital law to this path, we have
 

Therefore, B = 0.

This implies, magnetic field outside the solenoid is 0.
Field inside the solenoid:
Consider a closed path pqrs.
The line integral of magnetic field  is given by, 
 

For path pq,  and  are along the same direction,

For path rs, B = 0 because outside the solenoid field is zero.

Using these equations, equation (i) gives, 

Now, using Ampere’s law,
    

This implies,
      

b)
Magnetic lines do not exist outside the body of a toroid. Toroid is closed and solenoid is open on both sides. Magnetic field is uniform inside a toroid whereas, for a solenoid it is different at two ends and centre. 
   

c) The magnetic field is made strong by,

    i) passing large current and

   ii) using laminated coil of soft iron.