(a) A circular current carrying coil has a radius `R`. Find magnetic field (a) at centre and (b) along the axis of coil distant `sqrt(3) R` from centre. The coil is having `N` turns and carriers a current `i`.
(b) Two concentric coil `A` and `B`, having current `i` and `2i` and radii `2R` and `R` are placed as shown. Find magnetic field at common centre.

(c) In previous problem, if planes of coil are perpendicular to each other, find magnetic field at common centre.
(d) A charge `q` distributed uniformely over a circular ring of radius `R`. The ring rotates about its axis with an angular velocity `omega`. find the magnetic field (a) at centre and (b) at distance `sqrt(3)R` from centre, along the axis.

Explanation

(a) (i) Magnetic field at centre of coil
`B_(C)=(mu_(0)Ni)/(2R)`
(ii) Magnetic field at point `P`, on the axis of coil at distance `x` from centre
`B_(P)=(mu_(0)NiR^(2))/(2(R^(2)+x^(2))^(3//2))=(mu_(0)NiR^(2))/(2{R^(2)+(sqrt(3)R)^(2)}^(3//2))`
`=(mu_(0)NiR^(2))/(2(4R^(2))^(3//2))=(mu_(0)NiR^(2))/(2xx2sqrt(2)R^(3))=(mu_(0)Ni)/(4sqrt(2)R)`
(b)
At `O:`
`B_(1)=(mu_(0)(2I))/(2R)=(mu_(0)i)/R, o.`
`B_(2)=(mu_(0)i)/(2(2R))=(mu_(0)i)/(4R), ox`
`B_(O)=B_(1)-B_(2)=(2mu_(0)i)/(4R), o.`
(c) If planes of coils are `bot^(ar)`, `B_(1)` and `B_(2)` will be `bot^(ar)`

`B_(O)=(mu_(0)i)/R sqrt((1)^(2)+1/((4)^(2)))`
`=(sqrt(17)mu_(0)i)/(4R)`
(d) Here current `i=q/T=q/(2pi//omega)=(qomega)/(2pi)`
(i) `B_(C)=(mu_(0)i)/R=(mu_(0)q omega)/(4piR)`
(ii) `B_(P)=(mu_(0)iR^(2))/(2[R^(2)+(sqrt(3)R)^(2)]^(3//2))`
`=(mu_(0)iR^(2))/(4sqrt(2)R^(3))=(mu_(0)i)/(4sqrt(2)R)=(mu_(0)q omega)/(8sqrt(2) piR)`