Torque on a Current Loop  🔄

1 ️⃣ Rectangular loop in a uniform magnetic field

Imagine a rectangular coil of area \(A = ab\) carrying a steady current \(I\) and sitting in a region where the magnetic field \(\mathbf{B}\) is perfectly uniform. Two opposite sides feel no push, while the other two experience equal and opposite forces. These forces form a couple that tries to twist the coil. 💫:contentReference[oaicite:0]{index=0}

  • Magnitude of each force on the sides: \(F = I\,b\,B\) :contentReference[oaicite:1]{index=1}
  • Torque when the field lies in the plane of the coil:
    \[\tau = I\,A\,B\] :contentReference[oaicite:2]{index=2}
  • If the field makes an angle \(\theta\) with the coil’s normal:
    \[\tau = I\,A\,B\,\sin\theta\] 😊:contentReference[oaicite:3]{index=3}

Magnetic moment 🧭

  • \(\mathbf{m} = I\,\mathbf{A}\) (direction by the right-hand-thumb rule) :contentReference[oaicite:4]{index=4}
  • Vector form of the torque:
    \[\boldsymbol{\tau} = \mathbf{m} \times \mathbf{B}\] :contentReference[oaicite:5]{index=5}
  • If the coil has \(N\) turns: \(\mathbf{m} = N\,I\,\mathbf{A}\) ✨:contentReference[oaicite:6]{index=6}
  • Unit of \(\mathbf{m}\): ampere-metre2 (A m2)

Equilibrium ideas 🧘‍♂️

  • \(\tau = 0\) when \(\mathbf{m}\) is parallel or antiparallel to \(\mathbf{B}\).
  • Parallel → stable (a tiny nudge brings it back).
    Antiparallel → unstable (a nudge sends it spinning away).:contentReference[oaicite:7]{index=7}

2 ️⃣ Circular loop behaves like a magnetic dipole 🌐

For a loop of radius \(R\) carrying current \(I\), far from the loop (\(x \gg R\)) the field looks just like that of an electric dipole, but with magnetic quantities! 💡:contentReference[oaicite:8]{index=8}

  • Along the loop’s axis:
    \[B = \frac{\mu_0}{4\pi} \,\frac{2m}{x^{3}}\] where \(m = I\,A = I\,\pi R^{2}\). :contentReference[oaicite:9]{index=9}
  • On the perpendicular bisector of the loop:
    \[B = \frac{\mu_0}{4\pi} \,\frac{m}{x^{3}}\] (same distance limit). :contentReference[oaicite:10]{index=10}
  • No magnetic “charges” (monopoles) have ever been spotted, so the loop itself is the fundamental magnetic element. 🔍:contentReference[oaicite:11]{index=11}

3 ️⃣ Quick example snapshots ⚡

  • Earth’s field vs. ampere definition – A 1 A wire on a table feels a force per metre of \(3\times10^{-5}\,\text{N m}^{-1}\) in Earth’s field, much bigger than \(2\times10^{-7}\,\text{N m}^{-1}\) used in defining the ampere, so stray fields must be cancelled in precision setups. 🌍:contentReference[oaicite:12]{index=12}
  • 100-turn circular coil – For \(N=100,\; r=0.10\text{ m},\; I=3.2\text{ A}\):
      \(B_{\text{centre}} = 2\times10^{-3}\,\text{T}\),  \(m = 10\,\text{A m}^{2}\).
    In a 2 T field the torque jumps from 0 to 20 N m when the coil rotates 90°. 💪:contentReference[oaicite:13]{index=13}

🎯 High-Yield NEET Nuggets

  1. \(\boldsymbol{\tau} = \mathbf{m} \times \mathbf{B}\)  is the go-to formula for torque on any current loop.
  2. Magnetic moment shortcut: \(m = NIA\); remember its unit A m2.
  3. Stable equilibrium → \(\mathbf{m}\) parallel \(\mathbf{B}\); torque tries to align the dipole.
  4. A current loop is a magnetic dipole: far-field \(B \propto m/x^{3}\) (axis: factor 2).
  5. Torque vanishes when \(\theta = 0^\circ\) or \(180^\circ\); peaks at \(\theta = 90^\circ\).

Keep these ideas handy, practice with simple loops, and you’ll spin through magnetic-dipole questions with ease! ✨