Two Wires and a Compass: Superposition Under Constraint
Two long straight parallel wires run perpendicular to a tabletop, 0.30 m apart. Wire 1 carries current I₁ = 8.0 A into the table; wire 2 carries I₂ = 6.0 A out of the table. A small compass is placed at point P on the tabletop, on the line joining the wires, 0.10 m from wire 1 on the side away from wire 2. A student predicts the field at P points straight up the page because the currents are opposite and the fields 'cancel.' The instructor asks for a vector superposition and a check of the direction using the right-hand rule.
1. Compute the net magnetic field at P (magnitude and direction) and identify the flaw in the student's reasoning.
Read the worked answer
Let the line from wire 1 to wire 2 be the +x direction. At P (0.10 m from wire 1 on the −x side), the field from wire 1 (current into page) circles clockwise; at P the field points in the −y direction (down the page): B₁ = μ₀I₁/(2πr₁) = (2×10⁻⁷)(8.0)/0.10 = 1.6×10⁻⁵ T, −y. Wire 2 is 0.40 m from P; current out of page gives counterclockwise field, which at P points in the −y direction as well: B₂ = (2×10⁻⁷)(6.0)/0.40 = 3.0×10⁻⁶ T, −y. Net B = 1.9×10⁻⁵ T in the −y direction. The fields add, not cancel, because both wires produce downward fields at P; opposite currents do not imply opposite fields at a given point.
2. Is there a point on the line between the wires where the net field is zero? Justify with an equation.
Read the worked answer
Between the wires, the fields from the two wires point in opposite directions (wire 1's field points −y, wire 2's field points +y there), so cancellation is possible. Set μ₀I₁/(2πx) = μ₀I₂/(2π(0.30−x)) with x measured from wire 1: 8.0/x = 6.0/(0.30−x) → 8.0(0.30−x) = 6.0x → 2.4 = 14x → x ≈ 0.171 m from wire 1. So yes, at about 0.17 m from wire 1 (0.13 m from wire 2) the fields cancel. Outside the pair, on the side of the weaker wire, a second null point exists at x = 1.2 m from wire 1 (beyond wire 2), where 8.0/x = 6.0/(x−0.30).