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Ampère’s Law

4.3 Ampère’s Law

What is Ampère’s Law?
Ampère’s Law helps us calculate the magnetic field (\(\vec{B}\)) created around a wire when electric current flows through it. It gives a clear relation between the current flowing through a conductor and the magnetic field it creates around it.

Think of current as water flowing through a pipe, and the magnetic field as invisible ripples forming around the pipe — Ampère’s Law tells us how strong those ripples are and how they behave.

Before we dive in: What is a closed path?

Before understanding Ampère’s Law, you need to understand the idea of a closed path.

  • A closed path means any loop that starts and ends at the same point.
  • If we measure something along such a path, it's called a line integral over a closed loop.

For example, in a circular path around a wire, we can measure the magnetic field all around that circle. This is exactly what Ampère’s Law does — it looks at the magnetic field along that loop.

The mathematical form is:

\[ \oint \vec{B} \cdot d\vec{l} = B \, dl \cos\theta \]

Where:

  • \(\vec{B}\) = magnetic field at a point on the path
  • \(d\vec{l}\) = small length along the path
  • \(\theta\) = angle between \(\vec{B}\) and \(d\vec{l}\)

Statement of Ampère’s Law

"The total magnetic field around a closed loop is directly proportional to the total current enclosed by that loop."

In equation form:

\[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I \quad \text{(Eq. 4.19)} \]

Where:

  • \(\mu_0\) = magnetic permeability of free space
  • \(I\) = current enclosed by the path

Example to Understand

Suppose you have a straight long wire and a constant current \(I\) flows through it. This current creates circular magnetic field lines around the wire (like rings around a stick).

If you place a tiny magnetic needle near the wire, it will try to rotate — this shows that a magnetic field exists. The magnetic field applies a torque (twisting force) on the needle.

The torque on a magnetic dipole is given by:

\[ \tau = \vec{p} \times \vec{B} = pB \sin\theta \quad \text{(Eq. 4.20, 4.21)} \]

Where:

  • \(p\) = magnetic moment of the needle
  • \(B\) = magnetic field strength
  • \(\theta\) = angle between \(\vec{p}\) and \(\vec{B}\)

Deriving Magnetic Field Around a Straight Wire

Now let's apply Ampère’s Law to find magnetic field \(B\) at a distance \(r\) from a long straight wire carrying current \(I\).

From observation:

\[ B \propto \frac{I}{r} \quad \Rightarrow \quad B = \frac{\mu_0 I}{2\pi r} \quad \text{(Eq. 4.22)} \]

This means:

  • Magnetic field is stronger if current \(I\) increases
  • Magnetic field is weaker if distance \(r\) increases

Final Form of Ampère’s Law (Derivation)

We know the circumference of a circle is \(2\pi r\), and the magnetic field \(\vec{B}\) is tangent to every point on this circular path and has same magnitude at every point.

So,

\[ \oint \vec{B} \cdot d\vec{l} = B \cdot 2\pi r \]

Using Ampère’s Law:

\[ B \cdot 2\pi r = \mu_0 I \quad \Rightarrow \quad \boxed{B = \frac{\mu_0 I}{2\pi r}} \]

Again confirming:

\[ \boxed{\oint \vec{B} \cdot d\vec{l} = \mu_0 I} \quad \text{(Eq. 4.25)} \]

Summary

  • Ampère’s Law tells us how electric current creates magnetic fields.
  • It’s especially useful when the system has symmetry (e.g., long wires, solenoids, loops).
  • It’s one of the four fundamental Maxwell’s equations that govern electromagnetism.

Simple Analogy

Imagine throwing a stone in a pond. The ripples spread out in circles.
Now imagine current in a wire is like that stone — the current creates circular magnetic ripples around the wire.
Ampère’s Law helps you measure the strength of those ripples, depending on how strong the "stone" (current) was.

Let me know if you’d like this explained further in a diagram, in Bangla, or in PDF format.

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