Questions: Force Between Parallel Current-Carrying Wires

Question 1

Two parallel wires carry currents in opposite directions. A student, reasoning by analogy with electric charges, predicts the wires will attract. Is this correct?

• A. Yes — opposite currents attract, just as opposite charges attract
• B. No — parallel wires with opposite (antiparallel) currents repel each other
• C. No — opposite currents produce no force because their magnetic fields cancel
• D. Yes, but only if the wires are separated by less than one meter

Answer: (B)

Antiparallel currents repel — the opposite of what the charge analogy would suggest. Use the right-hand rule: if wire 1 carries current in +z and wire 2 carries current in −z, wire 1's field at wire 2's location still points in the same direction, but wire 2's current is reversed, so the force reverses. The charge analogy breaks down entirely here: unlike charges attract because their fields point toward each other; antiparallel currents repel because their interaction involves a cross product of current direction with magnetic field direction.

Question 2

Two parallel wires separated by distance d each carry current I. If the current in both wires is doubled and their separation is halved, by what factor does the force per unit length change?

• A. Doubles (×2)
• B. Quadruples (×4)
• C. Increases by a factor of 8 (×8)
• D. Increases by a factor of 16 (×16)

Answer: (C)

F/L = μ₀I₁I₂/(2πd). Doubling both currents multiplies I₁I₂ by 4. Halving the separation multiplies 1/d by 2. Combined effect: 4 × 2 = 8. The force grows with the product of the currents (so doubling both quadruples that factor) and inversely with distance (so halving the gap doubles that factor). The 1/d dependence is linear, not squared — an important distinction from Coulomb's law for point charges.

Question 3

Two parallel wires carrying current in the same direction will attract each other.

• T. True
• F. False

Answer: True

Parallel (same-direction) currents attract. Using the right-hand rule: wire 1's field at wire 2's location points perpendicular to the plane containing both wires. The force on wire 2 (F = IL × B) then points toward wire 1. This result — parallel currents attract — is counterintuitive relative to charge behavior but follows directly from the cross-product geometry. It was so fundamental that the original SI definition of the ampere was based on this force.

Question 4

The force between two parallel current-carrying wires follows an inverse-square law, falling off as 1/d².

• T. True
• F. False

Answer: False

The force per unit length falls off as 1/d (linear inverse), not 1/d² (inverse square). F/L = μ₀I₁I₂/(2πd). This is because the magnetic field from an infinite straight wire falls off as 1/d (by Biot-Savart or Ampère's law), and the force is proportional to that field. The 1/r² law applies to point charges (Coulomb's law) and point magnetic dipoles, not to infinite line sources. The 1/d behavior makes parallel wire interactions longer-range than point-charge interactions at comparable distances.

Question 5

Explain, using the right-hand rule, why parallel currents attract while antiparallel currents repel.

Model answer: Consider two wires both carrying current in the +z direction. By the right-hand rule, wire 1 creates a magnetic field circling counterclockwise around it. At wire 2's location (say, in the +x direction from wire 1), this field points in the −y direction. The force on wire 2 is F = I₂(ẑ) × (−ŷ) = I₂(x̂) — pointing toward wire 1, i.e., attraction. For antiparallel currents, wire 2 carries current in −z, so the force is I₂(−ẑ) × (−ŷ) = −I₂(x̂) — pointing away from wire 1, i.e., repulsion. The key is that the magnetic field direction is fixed by wire 1, but whether the resulting force is attractive or repulsive depends on wire 2's current direction via the cross product.

The right-hand rule for this problem has two applications: first to find the field from wire 1 (curl the fingers in the direction of current, field circles the wire), then to find the force on wire 2 using F = IL × B (point fingers in current direction, curl toward B, thumb points in force direction). The two-step application is what students often confuse, but working through it concretely with specific directions makes the attraction/repulsion result clear.