A current `I` flows along a round loop. Find the intergal `int B dr` along the axi so the loop within the range from `-oo` to `+oo`. Expain the result obtained.

Explanation

On the axis, `B = (mu_(0) I R^(2))/(2(R^(2) + x^(2))^(3//2)) = B_(x)`, along the axis.
Thus. `int vec(B) d vec(r) = int_(-oo)^(oo) B_(x) dx = (mu_(0) I R^(2))/(2) int_(-oo)^(oo) (dx)/((R^(2) + x^(2))^(3//2))`
`= (mu_(0) I R^(2))/(2) int_(-pi//2)^(pi//2) (R sec^(2) theta d theta)/(R^(3) sec^(3) theta)`, on putting `x = R tan theta`
`= mu_(0) I (1)/(2) int_(-pi//2)^(pi//2) cos theta d theta = mu_(0) I`
The physicall interpreation of this result is that `int_(-oo)^(oo) B_(x) dx` can be thought of as teh circulation of `B` over a closed loop by imaging that the two ends of the axis are connected by a line at inifinity (e.g, a semicircle of infinitate radius ).