Question

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?

Accepted Answer

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. 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=μ_{0}I2R, where μ_{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. 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. 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=μ_{0}I2r, where r is the distance from the wire to the point of interest. 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}=$$B_{loop}2$$+$$B_{wire}2. $$The direction of the net field can be determined by considering the vector components.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Magnetic Field Due to a Current Loop

Right-Hand Rule

Effect of Loop Orientation on Magnetic Field