Question

A current carrying wire is arranged at any angle in an uniform magnetic field, then

• A. only force acts on wire
• B. only torque acts on wire
• C. both
• D. none

Answer 1

Answer: (A)

only force acts on wire

Answer 2

When a current-carrying wire is placed in a uniform magnetic field, the magnetic force acting on a small segment of the wire $d\vec{l}$ is given by the Lorentz force:

$d\vec{F} = I (d\vec{l} \times \vec{B})$

For a straight current-carrying wire of length $L$ with current $I$ in a uniform magnetic field $\vec{B}$, the total magnetic force $\vec{F}$ on the wire is:

$\vec{F} = I (\vec{L} \times \vec{B})$

The magnitude of this force is $F = I L B \sin\theta$, where $\theta$ is the angle between the direction of the current (or the length vector $\vec{L}$) and the magnetic field $\vec{B}$.

1. Force: If the wire is arranged "at any angle", it implies that $\theta$ is not $0^\circ$ or $180^\circ$ (i.e., the wire is not parallel or anti-parallel to the magnetic field). In this case, $\sin\theta \neq 0$, and therefore, the force $F = I L B \sin\theta$ will be non-zero. So, a force acts on the wire.

2. Torque: Torque is the rotational effect of a force. For a rigid body (like a straight wire), if a net force acts through its center of mass, it causes pure translational motion. If the net force does not act through the center of mass, or if there are multiple forces forming a couple, then a net torque is produced, causing rotational motion.

For a straight current-carrying wire in a uniform magnetic field, the magnetic force $d\vec{F} = I (d\vec{l} \times \vec{B})$ on every segment $d\vec{l}$ of the wire is in the same direction (perpendicular to both $\vec{L}$ and $\vec{B}$). The resultant force $\vec{F} = I (\vec{L} \times \vec{B})$ acts through the center of mass of the wire (assuming uniform current and mass distribution). Since the force acts through the center of mass, it causes only translational motion of the wire and does not produce a net torque about the wire's center of mass.

A torque would be produced if:

• The wire was a loop (e.g., a current loop in a uniform magnetic field experiences a net torque but zero net force).
• The magnetic field was non-uniform.
• The wire was pivoted at a point other than its center of mass, in which case the force would create a torque about the pivot. However, the question asks what acts "on wire" itself, implying its intrinsic motion.

Therefore, for a straight current-carrying wire in a uniform magnetic field at an angle, only a net translational force acts on it.