Consider a circular current-carrying loop of radius R in the x-y plane with centre at origin. Consider the line integral `zeta(L)=|int_(-L)^(L)vecB.dvecl|` taken along z-axis.
(a) Show that `zeta(L)` monotonically increases with L.
(b) Use an appropriate Amperian loop to show that `zeta(oo)=mu_0I`, where I is the current in the wire.
(c) Verify directly the above result.
(d) Suppose we replace the circular coil by a square coil of sides R carrying the same current I. What can you say about `zeta(L)` and `zeta(oo)`?

Explanation

(a). B(z) points in the same direction on z-axis and hence , J(L) is a monotonically function of L. ltbtgt Since B and dl along the same direction, therefore B.dl=B.dl as cos0=1
(b). J(L)+contribution from large distance on contour `C=mu_(0)I` ,brgt `therefore` as `Lto infty`
Contribution form large distance `to0` (as `Bprop1//r^(3)`)
`J(infty)-mu_(0)I`
(c). the magnetic field due to circular current-carrying loop of radius R in the x-y plane with centre at origin at any point lying at a distance of from origin.
`B_(Z)=(mu_(0)lR^(2))/(2(Z^(2)+R^(2))^(3//2))`
`int_(-infty)^(infty)B_(Z)dz=int_(-infty)^(infty)(mu_(0)lR^(2))/(2(Z^(2)+R^(2))^(3//2))dz`
Put `Z=Rtantheta_(1)`
`impliesdZ=Rsec^(2)theta d theta`
`thereforeint_(-infty)^(infty)B_(z)dz=(mu_(0)I)/(2)int_(-pi//2)^(pi//2)costheta d theta=mu_(0)I`
(d). `B(Z)_("square")ltB(Z)_("circular coil")`
`sum(L)_("square")lt sum(L)_("circular coil")`
but by using arguments as in (b)
`sum(infty)_("square")=sun(infty)_("circular")`