A Wire, a Compass, and a Loop
A long straight horizontal wire runs north–south and carries 4.0 A toward the north. A small compass is placed 0.050 m directly below the wire. Nearby, a separate experiment has a circular loop of radius 0.10 m carrying 2.0 A; a student is asked about the magnetic field at the loop's center. A third setup involves a charged particle moving parallel to the straight wire. The student says: 'The compass below the wire will point north because the wire's field is along the wire.' Use μ₀ = 4π×10⁻⁷ T·m/A.
1. Compute the magnetic field magnitude and direction produced by the straight wire at the compass location, and evaluate the student's claim about where the compass points.
Read the worked answer
B = μ₀I/(2πr) = (4π×10⁻⁷)(4.0)/(2π×0.050) = (2×10⁻⁷)(4.0)/0.050 = 8×10⁻⁷/0.050 = 1.6×10⁻⁵ T. Direction: current is north; at a point below the wire, the right-hand rule (thumb north, fingers curl) gives field pointing east (or west depending on below/above; below the wire the field points east). Either way it is horizontal and perpendicular to the wire, not along it. The compass needle aligns with the net field (wire field plus Earth's ~5×10⁻⁵ T horizontal component), so it will not simply point north. The student's claim is wrong: the field circles the wire.
2. At the center of the circular loop, compute the field magnitude and state its direction relative to the loop's plane.
Read the worked answer
B = μ₀I/(2R) = (4π×10⁻⁷)(2.0)/(2×0.10) = (4π×10⁻⁷)(2.0)/0.20 = 2.51×10⁻⁶/0.20 ≈ 1.26×10⁻⁵ T. Direction: perpendicular to the plane of the loop, given by the right-hand rule (curl fingers in current direction, thumb points along B). This contrasts with the straight wire, where the field circles the wire; at the loop center the field is along the axis, not in the plane. Recognizing which geometry gives axial vs. circulating field is the judgment being tested.
3. The charged particle moves parallel to the straight wire at the same 4.0 A-equivalent drift. Explain whether it experiences a magnetic force, and how this differs from a charge moving perpendicular to the wire.
Read the worked answer
The wire's field at the particle's location circles the wire; a particle moving parallel to the wire moves along the wire axis, which is perpendicular to the circulating field lines at that point. Since F = qv×B and v ⊥ B, the force is nonzero (magnitude qvB, direction tangential/radial depending on side). The subtlety: 'moving parallel to the wire' does not mean 'moving parallel to B' - B is not along the wire. A charge moving exactly along a field line (e.g., along the loop's axis at its center) would feel zero magnetic force. The discriminator is the angle between v and B, not between v and the wire.