Question

A current-carrying wire is positioned within a large, uniform magnetic field, \(\vec{B}\). However, the wire experiences no force. Explain how this might be possible.

Short answer

Answer: A current-carrying wire will experience no force in a uniform magnetic field if there is no current flowing through the wire (I=0), if there is no magnetic field (B=0), or if the direction of the current is parallel or anti-parallel to the direction of the magnetic field (θ=0° or 180°).

Step-by-step solution

Understand the formula for magnetic force on a current-carrying wire

To find the conditions under which a current-carrying wire experiences no force in a magnetic field, let's recall the formula for the magnetic force acting on a current-carrying wire: \(F = I L B \sin(\theta)\) Here, \(F\) is the magnetic force, \(I\) is the current flowing through the wire, \(L\) is the length of the wire, \(B\) is the magnitude of the magnetic field, and \(\theta\) is the angle between the direction of the current and the direction of the magnetic field.

Analyze when the magnetic force is zero

From the formula, we can see that the magnetic force acting on the wire will be zero if one or more of the following conditions are true: 1. The current \(I = 0\). In this case, there is no current flowing through the wire, so no magnetic force will act on the wire. 2. The magnetic field \(B = 0\). If there is no magnetic field, no magnetic force will act on the wire. 3. The angle \(\theta = 0^\circ\) or \(180^\circ\). If the direction of the current is either parallel or anti-parallel to the direction of the magnetic field, the magnetic force acting on the wire will be zero.

Conclusion

A current-carrying wire can experience no force in a uniform magnetic field if there is no current flowing through the wire (\(I = 0\)), if there is no magnetic field (\(B = 0\)), or if the direction of the current is parallel or anti-parallel to the direction of the magnetic field (\(\theta = 0^\circ\) or \(180^\circ\)).