A current `I = 1.00` A circulates in a round thin-wire loop of radius `R = 100 mm`. Find the magentic induction
(a) at the centre of the loop,
(b) at the point lying on the axis of the loop at a disatnace `x = 100 mm` from its centre.

Explanation

(a) From the Biot - Savart law,
`d vec(B) = (mu_(0))/(4 pi) i (d vec(l) xx vec(r))/(r^(3))`, so
`dB = (mu_(0))/(4pi) i ((R d theta) R)/(R^(3)) (as d vec(l) _|_ vecK(r))`
From the symmetry
`B = int dB = int_(0)^(2pI) (mu_(0))/(4pi) (i)/(R) d theta = (mu_(0))/(2) (i)/(R) = 6.3 mu T`
(b) From Biolt-Savant's law :
`vec(B) = (mu_(0))/(4pi) (i)int (d vec(d) xx vec(r))/(r^(3))` (here `vec(r) = vec(R) + vec(x)`)
So, `vec(B) = (mu_(0))/(4pi)i[oint d vec(l) xx vec(R) + oint d vec(l) xx vec(x)]`
Since `vec(x)` is a constant vector and `|vec(R)|` is also constant
So, `oint d vec(l) xx vec(x) = (oint d vec(l)) xx vec(x) = 0` (because `oint d vec(l) = 0)`
and `oint d vec(l) xx vec(R) = oint R dl vec(n)`
`= vec(n) R oint dl = 2pi R^(2) vec(n)`
here `vec(n)` is a unit vector perpendicular to the plane conatining the current loop (Fig) and in the direction of `vec(x)`
Thus we get `vec(B) = (mu_(0))/(4pi) (2pi R^(2)i)/((x^(2) + R^(2))^(3//2)) vec(n)`