Magnetic Fields from Current-Carrying Wires: Configurations and Superposition

Q14. The figure shows four different sets of insulated wires that cross each other at right angles without actually making electrical contact. The magnitude of the current is the same in all the wires, and the directions of current flow are as indicated. For which (if any) configuration will the magnetic field at the center of the square formed by the wires be equal to zero?

Background

Topic: Magnetic Fields from Current-Carrying Wires

This question tests your understanding of how magnetic fields from multiple wires combine at a point, specifically at the center of a square formed by four wires carrying current in different directions.

Key Terms and Formulas

• Magnetic field from a straight wire: $B = \frac{\mu_0 I}{2\pi r}$, where $I$ is the current, $r$ is the distance from the wire, and $\mu_0$ is the permeability of free space.

• Superposition principle: The net magnetic field at a point is the vector sum of the fields produced by each wire.

• Right-hand rule: Used to determine the direction of the magnetic field around a current-carrying wire.

Step-by-Step Guidance

1. Examine each configuration (A, B, C, D) and note the direction of current in each wire. Use the right-hand rule to determine the direction of the magnetic field produced by each wire at the center of the square.

2. For each wire, point your thumb in the direction of the current and curl your fingers; your fingers indicate the direction of the magnetic field around the wire. At the center, consider the contribution from each wire.

3. Sum the magnetic field vectors from all four wires for each configuration. If the fields cancel each other out (i.e., the vector sum is zero), then the net field at the center is zero.

4. Compare the vector sums for all four configurations to determine which, if any, result in a net magnetic field of zero at the center.

Try solving on your own before revealing the answer!

Final Answer: E) The field is not equal to zero in any of these cases.

In each configuration, the magnetic fields from the wires do not cancel out completely at the center, so the net field is not zero.