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 centre of the solenoid due to this circular current. Integrate this expression under proper limits to find the magnetic field at the centre of the solenoid. (b) Verify that if `lgtgta`, the field tends to `(B=(mu_0)ni)` and if `agtgt1`, the field tends to` B= ((mu_0)nil/2a)` . Interpret these results.

Explanation

(a) Magnetic field at centre due to circular `
` loop`
`B = int dB`
`=int_(0)^1 ((mu_0a^2nidx)/(4pi[(a^2+(l-2x)^2 )]^(3//2))`
`=int_(0)^1 ((mu_0 nia^2 dx)/(4pia^3[1+((l-2x)/(a))^2] ^(3//2))`
`=((mu_0ni) /(4pi a)(int_(0)^1) (dx)/([1+((l-2x)/(a)^2)]^(3/2)`
` = ((mu_0 ni)/(4pia). ((4pi a )/(sqrt 1+(2a)/(l))) = (mu_0ni) /(sqrt (1+((2a)/(l)^2)))) `
(b) `When agtgtl `
`Then B = mu_(0)ni /2a.`