What is the magnitude of the magnetic force on a current carrying wire?

Question

BlackTom AI · Accepted Answer

Question: What is the magnitude of the magnetic force on a current carrying wire?

Option a: F = q v B sin θ
- This form resembles the magnetic force on a moving charge, F = q v × B, which yields magnitude F = q v B sin θ for a single charge. However, a current-carrying wire involves a continuous distribution of many charges moving with drift velocity, not a single charge q. The total force on the wire scales with current I and length L, not with a single charge q. Therefore this option does not correctly describe the force on a wire.

Option b: F = I L B sin θ
- This is the standard expression for the magnetic force on a straight wire of length L carrying current I in a magnetic field B with angle θ between the wire and the magnetic field. The derivation comes from integrating the force on each moving charge or using the dF = I dℓ × B form, which yields a magnitude F = I L B sin θ for uniform B and straight wire. This matches the well-known result and is consistent with the cross product behavior of force on current elements.

Option c: F = |q| v B sin θ
- This option incorrectly treats the force as depending on a single charge magnitude |q| and its velocity v, similar to the point-charge formula. In a current-carrying wire, many charges contribute, and the relevant quantity is the current I and length L, not a single charge magnitude. The presence of |q| here makes it inappropriate for the whole wire's magnetic force.

Option d: F = |q| v B cos θ
- This expression uses cos θ instead of sin θ and still references a single charge with magnitude |q| and velocity v. It also wrongly applies the cos factor, which would not correspond to the magnitude of the magnetic force from the Lorentz force F = q v × B, whose magnitude is q v B sin θ. For a current-carrying wire, this combination is not correct for the total force on the wire.

Summary: Option a misapplies a single-charge formula to a distributed current, option c misuses a single-charge model with wrong trigonometric factor, and option d also misuses the single-charge framework with the incorrect angle. The form that correctly captures the total force on a straight current-carrying wire in a magnetic field is F = I L B sin θ, which is option b.

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