11.5 Force and Torque on a Current Loop

Magnetic forces and torques on current loops are key concepts in electromagnetism. They explain how electric currents interact with magnetic fields, creating forces and rotational effects that are crucial in many applications.

Understanding these interactions helps us grasp the behavior of electric motors, generators, and other electromagnetic devices. The interplay between current loops and magnetic fields forms the basis for converting electrical energy into mechanical motion and vice versa.

Magnetic Forces and Torques on Current Loops

Net force on current loops

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  6.8 Torque on a Current Loop: Motors and Meters – Douglas College Physics 1207 View original

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  22.8 Torque on a Current Loop: Motors and Meters – College Physics View original

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  6.8 Torque on a Current Loop: Motors and Meters – Douglas College Physics 1207 View original

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• Sum of forces acting on each segment of the loop in an external magnetic field
• Force on each segment given by Lorentz force law: $\vec{F} = I\vec{L} \times \vec{B}$
  • $I$ represents current in the loop
  • $\vec{L}$ represents vector for length and direction of the segment
  • $\vec{B}$ represents external magnetic field
• Rectangular loop with sides of length $a$ and $b$ in a uniform magnetic field has zero net force
  • Forces on opposite sides of the loop cancel each other out (top and bottom, left and right)
• Non-uniform magnetic field may result in non-zero net force on the loop
  • Force depends on magnetic field gradient across the loop (change in field strength from one side to another)

Torque on current loops

• Current loop in a magnetic field experiences a torque that aligns the loop with the field
• Torque on a current loop given by: $\vec{\tau} = \vec{m} \times \vec{B}$
  • $\vec{m}$ represents magnetic dipole moment of the loop
  • $\vec{B}$ represents external magnetic field
• Magnitude of the torque given by: $\tau = mB\sin\theta$
  • $\theta$ represents angle between magnetic dipole moment and magnetic field
• Direction of torque is perpendicular to both magnetic dipole moment and magnetic field (determined by right-hand rule)
• Maximum torque when loop is perpendicular to magnetic field ($\theta = 90°$)
• Zero torque when loop is parallel to field ($\theta = 0°$ or $180°$)

Magnetic dipole moment concept

• Vector quantity $\vec{m}$ characterizes magnetic properties of a current loop
• Magnitude of magnetic dipole moment given by: $m = IA$
  • $I$ represents current in the loop
  • $A$ represents area enclosed by the loop
• Direction of magnetic dipole moment is perpendicular to plane of the loop (determined by right-hand rule)
  • If fingers of right hand curl in direction of current, thumb points in direction of magnetic dipole moment
• Analogous to electric dipole moment, but for magnetic fields instead of electric fields
• Interaction between current loop's magnetic dipole moment and external magnetic field responsible for torque experienced by the loop
• Potential energy of a magnetic dipole in a magnetic field given by: $U = -\vec{m} \cdot \vec{B}$
  • Minus sign indicates dipole tends to align itself with the field to minimize potential energy (stable equilibrium position)

Magnetic Field Sources and Interactions

• Ampere's law relates the magnetic field around a closed loop to the electric current passing through the loop
• Solenoid is a long coil of wire that generates a uniform magnetic field inside when current flows through it
• Magnetic flux is a measure of the total magnetic field passing through a given area
• Magnetic susceptibility describes how easily a material can be magnetized in response to an external magnetic field
• Larmor precession is the precession of the magnetic moments of electrons, atoms, or nuclei around the direction of an external magnetic field