§ 35. Magnetic Interaction of Parallel Currents & Definition of the Ampere (Grade 9–11)
Magnetic interaction of two parallel conductors carrying currents I₁ and I₂, attraction vs. repulsion. Two parallel conductors carrying electric currents in the same direction attract each other, and when carrying currents in opposite directions, repel.
Theory Summary & Scientific Mechanism
In 1820, André-Marie Ampère discovered the mechanical force between two parallel straight conductors carrying currents I₁ and I₂ separated by distance d. The interaction is mediated by the magnetic field: wire 1 generates concentric magnetic field lines of magnitude B₁ = (μ₀ I₁)/(2π d) at the location of wire 2 (right-hand grip rule).
Wire 2 of length L carrying current I₂ in field B₁ experiences the Ampère force F₁₂ = I₂ [L × B₁], yielding a mutual force of magnitude F = (μ₀ I₁ I₂ L)/(2π d), where μ₀ = 4π× 10⁻⁷ H/m is the vacuum permeability.
Evaluating the double cross product I₂ × (I₁ × r₁₂) proves that parallel currents in the same direction attract, whereas antiparallel currents repel! Historically, this exact configuration defined the SI base unit of electric current, the Ampere (2× 10⁻⁷ N/m at d = 1 m).
Everyday Analogy
Two current-carrying wires act like magnets without a single piece of iron! When currents flow in the same direction, the wires pull together; when currents flow in opposite directions, they push apart.
Interactive Question & Derivation
Two long parallel conductors separated by distance d carry currents I₁ and I₂ in the same direction (↑↑). How do the conductors interact, and how does the force per unit length F/l change if both currents are doubled?
- A: They attract each other, and the force per unit length F/l = (μ₀ I₁ I₂)/(2π d) increases 4 times
- B: They repel each other, and the force increases 2 times
- C: They do not interact magnetically unless the currents are alternating
Study Hint: By Ampère’s force law F = (μ₀ I₁ I₂ L)/(2π d): do parallel wires attract or repel when their currents flow in the same direction?
Proof & Explanation: By the Biot–Savart Law, current I₁ creates a circular magnetic field B₁ = (μ₀ I₁)/(2π d) at the location of the second conductor. Current I₂ in this field experiences the Ampère force F₁₂ = I₂[l×B₁] of magnitude (F)/(l) = (μ₀ I₁ I₂)/(2π d). By the right-hand and left-hand rules, parallel currents (↑↑) attract, whereas antiparallel currents (↑↓) repel. Doubling both I₁ and I₂ multiplies the product I₁ I₂, and thus the force, by 4!
Key Equations & 3D Simulation Parameters
(F)/(l) = (μ₀ I₁ I₂)/(2π d), B₁ = (μ₀ I₁)/(2π d)
- Left busbar current I₁: 5 … 30 A (default: 15 A)
- Right busbar current I₂: 5 … 30 A (default: 15 A)
- Conductor separation d: 1.2 … 3.4 cm (default: 2 cm)
- Mode (+1 parallel ↑↑, −1 antiparallel ↑↓): -1 … 1 (default: 1)