A tightly wound solenoid of radius 'a' and length 'l' has n turns per unit length. It carries an electric current i. Magnetic field at a distance `(l)/(4)` from one of the end (inside the solenoid on its axis) is `B=(mu_(0)ni(sqrt5+3))/(sqrtK)` for `l=4a`. Then find the value of K.

Answer

`40.00`

Explanation

Magnetic field inside the solenoid,
`B = ( mu _(0) n i )/( 2) [ sin (theta _(1)) + sin ( theta _(2)) ]`
`sin (theta _(1)) = ( 3a)/( sqrt (( 3 a) ^(2) + a ^(2))) = ( 3)/( sqrt10)`
`sin ( theta _(2)) = (a)/( sqrt ( a ^(2) + a ^(2))) = (1)/( sqrt2)`

`implies B = ( mu _(0) ni )/(2) [ (1)/(sqrt2) +(3)/(sqrt10) ]`
`implies B = ( mu _(0) ni ( sqrt5 + 3))/( sqrt40)`
`therefore K = 40`