A current I flows along a round loop. Find the integral `intvecB.vec(dr)` along the axis of the loop within the range from `(-oo)` to `(+oo)`. Explain the result obtained.

Explanation

The magnetic field at point P on the axis of a coil at distance
from the centre of the coil is
`B_x=(mu_0IR^2)/(2(R^2+x^@)^(3//2))`
the axis of the loop is taken along the x-axis.
`int_(-oo)^(+oo) vecB.vec(dr)=int_(-oo)^(+oo) B_xdx=(mu_0IR^2)/2 int_(-oo)^(+oo) (dx)/((R^2+x^2)^(3//2))`
`=(mu_0IR^2)/2 int_(-pi//2)^(pi//2) (Rsec^2 theta)/(R^3sec^3 theta)=(mu_0I)/2 int_(-pi//2)^(pi//2) cos theta d theta=mu_0I`

The integral on the LHS can be thought of as the line integral of
B over a closed path (a straight line from `(-oo)` to `(+oo)` and a semi-
circle of infinite radius where the magnetic field vanishes). Then
as per Amper's law, this must be `mu_0` times the current enclosed
by the path which is I.