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Two very long parallel wires carry steady currents I and 2I in opposite directions. The separation between the wires is d. At a given instant, a point charge q is located midway between the wires in the plane containing both wires. Its instantaneous velocity v is perpendicular to this plane. What is the magnitude of the magnetic force on the charge at this instant?
- mu₀ * I * q * v / (2 * pi * d)
- 3 * mu₀ * I * q * v / (pi * d)
- 3 * mu₀ * I * q * v / (2 * pi * d)
- Zero
Correct answer: 3 * mu₀ * I * q * v / (2 * pi * d)
Solution
The charge is midway between the wires, so distance from each wire = d/2. Magnetic field due to wire 1 (current I): B1 = mu₀*I/(2*pi*(d/2)) = mu₀*I/(pi*d). Magnetic field due to wire 2 (current 2I): B2 = mu₀*(2I)/(2*pi*(d/2)) = 2*mu₀*I/(pi*d). Since the currents are in opposite directions, using right-hand rule: both B1 and B2 point in the same direction at the midpoint (both pointing into or out of the plane containing the wires and the charge - but perpendicular to the velocity v). Wait: the velocity v is perpendicular to the plane of the wires. Magnetic fields from the wires are in the plane of the wires (perpendicular to each wire, lying in the plane). So B1 and B2 are in the plane of the wires. The force F = q(v x B) will be in the plane of the wires. Direction of B at midpoint: for two wires carrying opposite currents, B from wire 1 at midpoint and B from wire 2 at midpoint may add or cancel depending on geometry. If current I goes up in wire 1 and 2I goes down in wire 2 (or vice versa), using right-hand rule, B1 at the midpoint points in one direction (say, into the page of the wire plane) and B2 at the midpoint points in the SAME direction (because reversing current direction reverses B, but the position of the midpoint is on the other side for wire 2 - net effect depends on exact geometry). For parallel wires with opposite currents (like a transmission line), the fields between the wires add. B_total = B1 + B2 = mu₀*I/(pi*d) + 2*mu₀*I/(pi*d) = 3*mu₀*I/(pi*d). Force: F = q*v*B = 3*mu₀*I*q*v/(pi*d). This matches option B (3*mu₀*I*q*v/(pi*d)), not option C. Let me re-examine: option C is 3*mu₀*I*q*v/(2*pi*d) and option B is 3*mu₀*I*q*v/(pi*d). My calculation gives 3*mu₀*I*q*v/(pi*d). So the answer should be option B. But let me double-check the field formula: B = mu₀*I/(2*pi*r). For r = d/2: B1 = mu₀*I/(2*pi*(d/2)) = mu₀*I/(pi*d). B2 = mu₀*(2I)/(2*pi*(d/2)) = 2*mu₀*I/(pi*d). Total B = 3*mu₀*I/(pi*d). F = qvB = 3*mu₀*I*q*v/(pi*d). This matches option B: 3*mu₀*I*q*v/(pi*d).
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