"(II) A rectangular loop of wire is placed next to a straight wire, as shown in Fig. 28–40. There is a dc current of 3.5 A in both wires. Determine the magnitude and direction of the net force on the loop.

Verified step by step guidance

1. Step 1: Understand the problem setup. The rectangular loop of wire is placed next to a straight wire, and both carry a DC current of 3.5 A. The goal is to calculate the net force on the loop due to the magnetic interaction between the two wires. The magnetic force arises because the current in the straight wire generates a magnetic field, which interacts with the current in the loop.
2. Step 2: Recall the formula for the magnetic field generated by a straight wire. The magnetic field at a distance \( r \) from a long straight wire carrying current \( I \) is given by \( B = \frac{\mu_0 I}{2 \pi r} \), where \( \mu_0 \) is the permeability of free space (\( \mu_0 = 4 \pi \times 10^{-7} \; \text{T·m/A} \)). Use this formula to calculate the magnetic field at the location of each segment of the loop.
3. Step 3: Analyze the forces on each segment of the loop. The loop has four sides: two parallel to the straight wire and two perpendicular. For the parallel sides, the magnetic field from the straight wire interacts with the current in the loop, producing forces. Use the formula for the magnetic force on a current-carrying wire: \( F = I L B \), where \( I \) is the current, \( L \) is the length of the wire segment, and \( B \) is the magnetic field. Determine the direction of the forces using the right-hand rule.
4. Step 4: Consider the forces on the perpendicular sides of the loop. The magnetic field from the straight wire is parallel to these sides, so the force on these segments will be zero because the angle between the current direction and the magnetic field is 0° (\( F = I L B \sin \theta \), and \( \sin 0° = 0 \)).
5. Step 5: Calculate the net force on the loop. Add the forces from the two parallel sides, taking into account their magnitudes and directions. The forces may partially or completely cancel depending on the relative positions of the loop and the straight wire. Use vector addition to determine the net force and its direction.

Key Concepts

Magnetic Field Due to a Current-Carrying Wire

A straight wire carrying an electric current generates a magnetic field around it, described by the right-hand rule. The direction of the magnetic field lines forms concentric circles around the wire, with the strength of the field decreasing with distance from the wire. This magnetic field interacts with other currents, such as those in nearby loops, influencing their motion.

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving through a magnetic field. For a current-carrying conductor, this force can be calculated using the formula F = I(L × B), where I is the current, L is the length vector of the conductor, and B is the magnetic field. The direction of the force is determined by the right-hand rule, which helps predict how the loop will move in response to the magnetic field created by the straight wire.

Superposition of Forces

In physics, the principle of superposition states that when multiple forces act on an object, the total force is the vector sum of all individual forces. In the context of the rectangular loop near the straight wire, the net force on the loop is determined by considering the magnetic forces acting on each segment of the loop due to the magnetic field from the straight wire. This requires analyzing the contributions from each segment to find the overall effect on the loop.