A tightly-wound solenoid of radius a and length l has n turns per unit length and carries an electric current i . What does the magnetic field at the centre of the solenoid tend to in the limits l a and a l , respectively?

Step-by-step solution

The magnetic field at the centre of a finite solenoid of length l and radius a having n turns per unit length and carrying current i is given by B = ₀ n i 2 ( ₁ + ₂) where ₁ and ₂ are the semi-vertical angles subtended by the ends at the centre. At the centre of the solenoid, ₁ = ₂ = . From the geometry of the solenoid, = l 2 a^2 + ( l 2 )^2 = l 4a^2 + l^2 Substituting this into the expression for B , B = ₀ n i 2 ( 2l 4a^2 + l^2 ) = ₀ n i l 4a^2 + l^2 In the limit l a , the term 4a^2 can be neglected compared to l^