For a circular coil of radius R and N turns carrying current. Prove that the magnitude of the magnetic field at a point on its axis at a distance X from its centre is given by `B=(mu_0IR^2N)/(2(x^2+R^2)^(3//2))`

Explanation

(a) We have,
`B=(mu_(0)NIR^(2))/(2(x^(2)+R^(2))^(3/2))`
For finding magnetic field on the centre of above circular coil we should take x = 0 and then,
`B_("centre")=(mu_(0)NIR^(2))/(2R^(3))=(mu_(0)NI)/(2R)`
Above is also the magnetic field at the centre of current carrying extremely thin circular ring.
(b)
(1) Above is the figure as per the statement.
(2) Here, `PP_(1)=PP_(2)=DeltaxltltltltR`
(3) At point `P_(1)`, magnitude field due to coil-1 is,
`B_(1)=(mu_(0)NIR^(2))/(2(x_(1)^(2)+R^(2))^(3/2))`
`thereforeB_(1)=(mu_(0)NIR^(2))/(2{(R/2-Deltax)^(2)+R^(2)}^(3/2))" "...(1)`
`B_(1)=(mu_(0)NIR^(2))/(2{R^(2)/4+(Deltax)R+(Deltax)^(2)+R^(2)}^(3/2))`
(4) Now, since `Deltax` is very small, neglecting `(Deltax)Rand(Deltax)^(2)` we get,
`B_(1)~~(mu_(0)NIR^(2))/(2(5/4R^(2))^(3/2))=(mu_(0)NI)/(2(5/4)^(3/2)R)" "...(2)`
(5) At point `P_(1)`, magnetic field due to coil-2 is,
`B_(2)=(mu_(0)NIR^(2))/(2(x_(2)^(2)+R^(2))^(3/2))`
`thereforeB_(2)=(mu_(0)NIR^(2))/(2{(R/2+Deltax)^(2)+R^(2)}^(3/2))" "...(3)`
`thereforeB_(2)=(mu_(0)NIR^(2))/(2{R^(2)/4+(Deltax)R+(Deltax)^(2)+R^(2)}^(3/2))`
(6) Now since `Deltax` is very small, neglecting `(Deltax)Rand(Deltax)^(2)` we get,
`B_(2)~~(mu_(0)NIR^(2))/(2(5/4R^(2))^(3/2))=(mu_(0)NI)/(2(5/4)^(3/2)R)" "...(4)`
(7) Now, since `vecB_(1)||vecB_(2)`, resultant magnetic field at point `P_(1)` will be,
`B=B_(1)+B_(2)`
= `(mu_(0)NI)/((5/4)^(3/2)R)`
= `(4/5)^(3/2)(mu_(0)NI)/R`
= `=0.715(mu_(0)NI)/R`
`thereforeB~~0.72((mu_(0)NI)/R)" "...(5)`
(8) Similarly if we find out resultant magnetic field at point `P_(2)`, in above figure, we get it same as given be equation (5). Thus in above situation in the length `P_(1)P_(2)=2(Deltax)` along the common axis around centre point P, magnetic field remains uniform whose constant magnitude is equal to `0.72((mu_(0)NI)/R)`. Such a pair of coil (shown in the diagram) is called "Helmholtz coils".