🚀 Ampere’s Circuital Law

1. What the law says

Curl your right-hand fingers around any closed path that loops around a current. The path integral of the tangential magnetic field along that loop equals the permeability of free space times the total current passing through the surface it bounds: \( \displaystyle \oint \mathbf{B}\!\cdot\!d\mathbf{l}= \mu_0 I \) [Eq. 4.13(a)] :contentReference[oaicite:0]{index=0} Thumb ➡️ current direction, curled fingers ➡️ direction to walk along the loop.

2. Picking a smart “Amperian” loop

• Choose the loop so that at each point either (i) \( \mathbf{B} \) is tangential and constant, (ii) \( \mathbf{B} \) is ⟂ to the loop, or (iii) \( \mathbf{B}=0 \). :contentReference[oaicite:1]{index=1}
• If the loop has length \(L\) where the field stays tangential and equal to \(B\), the law turns into the super-simple formula \( BL=\mu_0 I_{\text{enclosed}} \). [Eq. 4.13(b)] :contentReference[oaicite:2]{index=2}
• This trick mirrors how Gauss’s law simplifies electric-field problems. 💡

3. Long straight wire 🌟 (classic example)

Wrap the loop as a circle of radius \(r\) centered on an infinitely long wire. Because symmetry makes \(B\) constant on the circle and tangential everywhere, you get \( B(2\pi r)=\mu_0 I \) ⇒ \( \displaystyle B=\frac{\mu_0 I}{2\pi r} \). [Eq. 4.14] :contentReference[oaicite:3]{index=3}

Key observations: 🌀 The field forms concentric circles (cylindrical symmetry). ⬇️ Magnitude falls off as \(1/r\). ✅ Direction by right-hand rule—thumb = current, fingers = field. :contentReference[oaicite:4]{index=4}

4. Solid wire with uniform current

For a wire of radius \(a\) carrying total current \(I\):

• Outside the wire \((r>a)\): same result as above \( \displaystyle B=\frac{\mu_0 I}{2\pi r}. \) :contentReference[oaicite:5]{index=5}
• Inside the wire \((r

The graph of \(B\) versus \(r\) rises linearly inside and drops as \(1/r\) outside. :contentReference[oaicite:7]{index=7}

5. Why the law rocks 😊

• Gives magnetic fields quickly when symmetry is present—no heavy calculus.
• Unites magnetism and current: change the current, and the loop integral changes instantly. :contentReference[oaicite:8]{index=8}
• Stays valid only for steady (time-independent) currents.

📌 High-Yield NEET Nuggets

1. Memorize the core formula \( \displaystyle \oint \mathbf{B}\!\cdot\!d\mathbf{l}= \mu_0 I \) and the right-hand sign convention. 🖐️ :contentReference[oaicite:9]{index=9}
2. Know the straight-wire field \( B=\mu_0 I / (2\pi r) \) and that it falls as \(1/r\). :contentReference[oaicite:10]{index=10}
3. Inside a uniformly charged (oops—current-carrying!) wire, \( B\propto r \); outside, \( B\propto 1/r \). :contentReference[oaicite:11]{index=11}
4. Right-hand rule: thumb = \(I\), fingers = \(\mathbf{B}\). Easy way to nail direction questions. :contentReference[oaicite:12]{index=12}
5. Symmetry choice of Amperian loop often turns tough integrals into one-liners—spot the symmetry! :contentReference[oaicite:13]{index=13}

✨ Keep practicing with different loops, and this law will feel like magic! ✨