Q. 57

A tightly-wound solenoid of radius a and length l has n turns per unit length. It carries an electric current i. Consider a length dx of the solenoid at a distance x from one end. This contains n dx turns and may be approximated as a circular current i n dx.

(a) Write the magnetic field at the center of the solenoid due to this circular current. Integrate this expression under proper limits to find the magnetic field at the center of the solenoid.

(b) Verify that if l >> a, the field tends to B = μ0ni and if a >> l, the field tends to

B=. Interpret these results

Answer :

Given


Radius of solenoid is a


Length of the solenoid is l


Number of turns per unit length is n


The current in the circular loop is indx


The magnetic field due to the circular ring at the any distance from the ring is



Where


μo is the permeability of the free space,


I is the current in the ring


r is the radius of the loop


x is the distance from the loop at which the magnetic field to be found.


Integrating the above equation for x-0 and r




On solving the above equation, we get



The magnetic field at the center of the solenoid due to this circular current



(b) When la


Putting in the equation (i), we get



Putting the values in the above formula, we get


B=μ0 n I


When a l


Putting in the equation (i), we get



Putting the above value in the equation (i), we get



Hence, proved if l >> a, the field tends to B = μ0ni and


if a >> l, the field tends to B, .


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