🚀 Force Between Two Parallel Currents

Two long, straight conductors a and b carry steady currents \(I_a\) and \(I_b\) and are separated by a distance \(d\). Each current creates a magnetic field that reaches the other wire and pushes or pulls on it.💡

1 🔧 Magnetic field from one wire

For wire a, the field at wire b has magnitude \[ B_a=\frac{\mu_0 I_a}{2\pi d}. \] :contentReference[oaicite:0]{index=0} The right-hand rule points the field “downwards” if the wires lie horizontally.

2 ⚡ Force on the second wire

Wire b (length \(L\)) feels a sideways force \[ F_{ba}=I_b\,L\,B_a =\frac{\mu_0 I_a I_b L}{2\pi d}\tag{4.17}. \] :contentReference[oaicite:1]{index=1} Wire a experiences an equal and opposite force: \[ F_{ab}=-F_{ba}. \] This matches Newton’s third law. 🎯

3 🤝 Attraction vs. Repulsion

Currents in the same direction attract; currents in opposite directions repel. “Like currents attract” is the magnetic opposite of “like charges repel.” :contentReference[oaicite:2]{index=2}

4 📏 Force per unit length

Dividing by \(L\) gives the handy form \[ f_{ba}=\frac{F_{ba}}{L} =\frac{\mu_0 I_a I_b}{2\pi d}\tag{4.19}, \] :contentReference[oaicite:3]{index=3} perfect for quick calculations and exam questions.

5 🔬 The ampere (A)

Imagine two very long, straight, parallel conductors 1 m apart in vacuum. If each carries a steady current that makes a force of \(2\times10^{-7}\,\text{N m}^{-1}\) on the other, that current is defined as 1 ampere. From this, \(1\) coulomb is the charge that passes a point in \(1\) s when the current is \(1\) A. :contentReference[oaicite:4]{index=4}

6 🌍 Quick check: Earth’s field example

  • Horizontal component of Earth’s field: \(B=3.0\times10^{-5}\,\text{T}\).
  • A 1 A wire placed east–west feels \(f = I B = 3\times10^{-5}\,\text{N m}^{-1}\) (force points downward).
  • The same wire placed south–north feels no force because \(\theta=0^\circ\). 👌

Engineers cancel Earth’s field with shielding coils when calibrating a current balance.:contentReference[oaicite:5]{index=5}

🎯 High-Yield Ideas for NEET

  1. \(B = \mu_0 I / 2\pi d\) for a long straight conductor.
  2. Force per unit length between two parallel currents: \(f = \mu_0 I_a I_b / 2\pi d\).
  3. Parallel currents attract; antiparallel currents repel.
  4. Operational definition of the ampere via magnetic force (\(2\times10^{-7}\,\text{N m}^{-1}\)).
  5. Link between ampere and coulomb: \(1\text{ C}=1\text{ A}\times1\text{ s}\).

Keep practicing ✨—solving a few force-between-wires questions each day builds confidence fast!