28.10
磁界内の電流ループのトルク
通電ワイヤに磁力を適用する最も一般的な用途は、電気モーターです。 これらは、磁場のある磁石の間に配置されたワイヤーのループで構成されています。 ループに電流が流れると、磁場によってトルクが発生し、シャフトが回転して、電気エネルギーが機械エネルギーに変換されます。
均一な磁場内に配置された、N 回のワ…
Consider a rectangular current-carrying loop of lengths a and b placed in a uniform magnetic field, with an axis of rotation passing through point O at distance m from one end.
The magnetic forces acting along the plane of the loop lie on the same axis passing through O; thus, the sum of their torques about the axis is equal to zero.
Similarly, the torques due to the forces perpendicular to the plane of the loop can be determined.
By adding all the torques acting on the loop, the net torque can be determined, where A is the area of the loop.
A current-carrying closed loop can be referred to as a magnetic dipole. The magnetic dipole moment is a vector quantity, where the magnitude is the product of the area and the current flowing through the loop; its direction is perpendicular to the plane of the loop, following the right-hand rule.
Therefore, the torque on a current loop due to a uniform magnetic field can be described in terms of the magnetic dipole moment.
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Q1: What is a magnetic dipole moment and how does it relate to a current loop?
A magnetic dipole moment is a vector quantity describing a current-carrying closed loop's magnetic properties. Its magnitude equals the product of the loop's area and current flowing through it, while its direction is perpendicular to the loop's plane, following the right-hand rule. This concept allows torque on current loops to be expressed in terms of the magnetic dipole moment.
Q2: How is torque calculated on a current-carrying loop in a magnetic field?
Torque on a current loop is calculated using the equation where I is current, A is loop area, B is magnetic field strength, and θ is the angle between current and field. Maximum torque occurs when θ = 90°, making sin θ = 1. Minimum torque occurs when θ = 0°, indicating the loop is aligned with the magnetic field.
Q3: Why is the net force on a current loop in a uniform magnetic field zero?
In a uniform magnetic field, magnetic forces acting along the loop's plane lie on the same rotation axis, causing their torques to cancel. Similarly, torques from forces perpendicular to the plane balance each other. This symmetry results in zero net force, though net torque can still exist depending on loop orientation.
Q4: What role does torque play in electric motor operation?
Electric motors use torque from magnetic forces on current-carrying wire loops to produce rotational motion. When current flows through loops placed between magnets, the magnetic field applies torque to the shaft, converting electrical energy into mechanical energy. This principle is fundamental to motor design and operation.
Q5: How does the angle between current and magnetic field affect torque magnitude?
Torque magnitude depends on sin θ, where θ is the angle between current direction and magnetic field. When θ = 90°, sin θ = 1 and torque is maximum. When θ = 0°, sin θ = 0 and torque is minimum. This angular dependence allows control of torque by adjusting loop orientation relative to the field.
Q6: How do you determine maximum torque on a multi-turn current loop?
For a multi-turn loop, multiply the single-turn torque equation by the number of turns N. For example, a 200-turn square loop with area 0.04 m² carrying 15 A in a 1-T field produces maximum torque of 120 N·m when perpendicular to the field. The torque equation becomes τ = NIAB sin θ.
Q7: What determines the direction of torque on a current loop in a magnetic field?
The torque direction is perpendicular to both the magnetic dipole moment vector and the magnetic field, following the right-hand rule. The magnetic dipole moment points perpendicular to the loop's plane in the direction determined by curling fingers with current flow. This vector relationship defines the rotational axis and direction of the loop's motion.