Wire orientation for maximum and zero force

Problem

A straight current-carrying wire is immersed in a uniform magnetic field. How would you orient the wire to obtain (a) the maximum force, (b) zero force? And if I double both the current and the field, by what factor does the force change? Argue without exact calculations.

Solution

The Laplace force on a straight segment of length $L$ has magnitude $F = i\,L\,B\,\sin\theta$ where $\theta$ is the angle between the wire and the field.

(a) Maximum force: obtained with the wire perpendicular to $\vec B$, i.e. $\theta = 90^\circ$ and $\sin\theta = 1$.

(b) Zero force: obtained with the wire parallel to $\vec B$, i.e. $\theta = 0$ and $\sin\theta = 0$.

Doubling both the current and the field at the same time, the force becomes $F' = (2i)(2B)\,L\sin\theta = 4\,F$: it increases by a factor of 4. $\ev{\text{(a) }\perp\vec B;\ \text{(b) }\parallel\vec B;\ \text{factor }4}$

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Topics: Magnetism Concepts: Laplace force Skills: Limiting-case and thought-experiment analysis