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

To solve the problem step-by-step, we will address each part of the question sequentially.

### Part (a): Show that `ζ(L)` monotonically increases with `L`.

1. **Understanding the Magnetic Field**: The magnetic field `B` due to a circular current-carrying loop at a point along the z-axis can be derived using the Biot-Savart law. The magnetic field at a distance `z` from the center of the loop is given by:
\[ B(z) = \frac{\mu_0 I R^2}{2(R^2 + z^2)^{3/2}} \]
...