Part of a long, thin insulated straight wire is formed into a single circular loop of radius 𝑅 (Fig. 28–68) and carries a current I. (a) What is the magnitude and direction of the magnetic field at the center of the loop? (b) If the plane of the loop is twisted 90 degrees so that the plane is perpendicular to the straight part of the wire (i.e., in the yz plane) what is the magnitude and direction of the field now at the center of the loop?

Verified step by step guidance

1. Step 1: Understand the problem and identify the relevant concepts. The problem involves calculating the magnetic field at the center of a circular loop of wire carrying current I. This requires using the Biot-Savart law or Ampere's law for part (a). For part (b), the orientation of the loop changes, and we need to analyze the new configuration to determine the magnetic field.
2. Step 2: For part (a), recall the formula for the magnetic field at the center of a circular loop carrying current I. The magnitude of the magnetic field is given by: $B=\frac{\mu }{}$₀I2R, where $\mu$₀ is the permeability of free space, I is the current, and R is the radius of the loop. The direction of the magnetic field can be determined using the right-hand rule.
3. Step 3: For part (b), when the plane of the loop is twisted 90 degrees, the loop is now perpendicular to the straight wire. The magnetic field at the center of the loop will be influenced by both the loop and the straight wire. Analyze the contributions from each component separately. The straight wire produces a magnetic field at the center of the loop, which can be calculated using the Biot-Savart law or Ampere's law.
4. Step 4: Combine the magnetic field contributions from the loop and the straight wire for part (b). The magnetic field from the loop remains the same in magnitude but changes direction due to the new orientation. The magnetic field from the straight wire depends on the distance from the wire to the center of the loop and follows the formula: $B=\frac{\mu }{}$₀I2r, where r is the distance from the wire to the point of interest.
5. Step 5: Use vector addition to determine the net magnetic field at the center of the loop for part (b). Since the magnetic fields from the loop and the straight wire are perpendicular to each other, their magnitudes can be combined using the Pythagorean theorem: $B$_net=B_loop2+B_wire2. The direction of the net field can be determined by considering the vector components.

Key Concepts

Magnetic Field Due to a Current Loop

The magnetic field at the center of a circular loop carrying a current can be determined using the Biot-Savart Law. The magnitude of the magnetic field (B) at the center is given by the formula B = (μ₀I)/(2R), where μ₀ is the permeability of free space, I is the current, and R is the radius of the loop. The direction of the magnetic field follows the right-hand rule, where the thumb points in the direction of the current, and the curled fingers indicate the direction of the magnetic field.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field around a current-carrying conductor. For a straight wire, if you point your thumb in the direction of the current, your fingers will curl in the direction of the magnetic field lines. This rule is essential for visualizing how the magnetic field orientation changes when the configuration of the current-carrying wire is altered, such as twisting the loop.

Effect of Loop Orientation on Magnetic Field

The orientation of a current-carrying loop affects the direction of the magnetic field it produces. When the plane of the loop is twisted, the magnetic field direction can change significantly. For instance, if the loop is oriented perpendicular to the straight wire, the magnetic field at the center will still be determined by the right-hand rule, but its direction will be different compared to when the loop is in the original plane.