Q.(a) State and prove Ampere's circuital law.

The Intuition: What Ampere's Law Is Really Saying

Imagine you're standing in a field of grass, and you walk in a complete circle. If the grass is perfectly flat and still, your path feels the same all the way around. But if there's a strong wind blowing through the center of your circle, you'll feel it push you differently at different points along your walk.

Electric currents create magnetic fields. Ampere's Circuital Law is a way of measuring how much magnetic field is swirling around a current — like measuring how strong the "whirlpool" of field lines is around a wire.

The key idea: if you take a closed loop (any shape you like) and add up the magnetic field along every tiny piece of that loop, the total you get is directly proportional to the amount of current that passes through the loop. No current through the loop? The total is zero.

The Precise Statement

Ampere's Circuital Law states:

Let's break down every symbol:

• $\oint$ — The circle on the integral sign means you're integrating around a closed loop. You start at some point, trace a complete path, and return to where you began.
• $\vec{B}$ — The magnetic field at each point on your loop.
• $d\vec{l}$ — An infinitesimally small piece of your loop, treated as a vector pointing along the direction you're walking.
• $\vec{B} \cdot d\vec{l}$ — The dot product. This picks up only the part of the magnetic field that points along your path. If the field is perpendicular to your path at some point, that piece contributes nothing.
• $\mu_0$ — The permeability of free space, a fundamental constant ($4\pi \times 10^{-7} \, \text{T·m/A}$). It tells you how "strongly" a current creates a magnetic field in empty space.
• $I_{\text{enc}}$ — The net current passing through the area bounded by your loop. "Net" means you add currents going one way and subtract currents going the opposite way.

Why the Dot Product Matters

The dot product $\vec{B} \cdot d\vec{l} = B \, dl \cos\theta$ where $\theta$ is the angle between the field and your path. This is crucial: if you walk along a path where the magnetic field is always perpendicular to your direction, you get zero contribution at every step — even if the field is huge.

This is why Ampere's Law is most useful for symmetric situations. You choose your loop so that:

1. The magnetic field is constant in magnitude along the loop.
2. The field is always parallel (or antiparallel) to your path, so $\cos\theta = \pm 1$.

Then the integral becomes simple multiplication: $B \times (\text{circumference of loop}) = \mu_0 I_{\text{enc}}$.

The Classic Example: A Straight Wire

Consider an infinitely long, straight wire carrying current $I$. The magnetic field circles around the wire in concentric circles. Choose your Amperian loop to be a circle of radius $r$ centered on the wire.

By symmetry, $B$ is the same at every point on the circle and points tangent to it — exactly along $d\vec{l}$. So:

Therefore:

This is the familiar formula for the field around a long straight wire. Notice: the field falls off as $1/r$, not $1/r^2$ like the electric field from a point charge. Magnetic fields from currents have a different geometry.