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

In order for a current-carrying wire within a uniform magnetic field to experience no magnetic force, the angle between the magnetic field's direction and the current's direction must be either 0 degrees (aligned in the same direction) or 180 degrees (aligned in opposite directions). It is under these conditions that the sine of the angle becomes zero, resulting in a magnetic force of zero as well.

Step-by-step solution

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

The magnetic force on a current-carrying wire can be calculated using the formula: F = ILBsin(θ) where F is the magnetic force, I is the current flowing through the wire, L is the wire's length, B is the magnetic field's strength, and θ is the angle between the magnetic field and the current's direction.

Analyze the terms contributing to magnetic force

For the wire to experience no magnetic force, the formula F = ILBsin(θ) must be equal to zero. This can happen if any of the following conditions are met: 1. The strength of the magnetic field (B) is zero. 2. No current flows through the wire (I = 0). 3. The length of the wire (L) is zero. 4. The angle between the magnetic field and the current's direction (θ) is such that sin(θ) = 0.

Determine the possible scenario for no magnetic force

Since the exercise states that the magnetic field is uniform and the wire is carrying a current, that means B ≠ 0 and I ≠ 0. A length of zero for the wire makes no physical sense, so L ≠ 0 also. That leaves us with the possibility that sin(θ) = 0.

Explore when sin(θ) = 0

The sine function equals zero at two possible angles: θ = 0 degrees and θ = 180 degrees. If θ = 0 degrees, the magnetic field and the current flow are aligned in the same direction. If θ = 180 degrees, the magnetic field and current flow are in opposite directions.

Explain the scenario for no magnetic force

The wire experiences no magnetic force if the angle between the direction of the magnetic field and the direction of current flow is either 0 degrees (aligned in the same direction) or 180 degrees (aligned in opposite directions). In both cases, the sine of the angle is zero, which results in a magnetic force of zero.