Electric motors harness the power of magnetism to convert electrical energy into mechanical work. By placing a current-carrying loop in a magnetic field, a torque is generated, causing rotation. This fundamental principle underlies the operation of countless devices we use daily.

The interaction between current and magnetic field in motors is governed by key equations like the Lorentz force law and torque formula. Understanding these relationships helps explain how motors achieve continuous rotation through clever design elements like brushes and commutators.

Torque on a Current Loop: Motors and Meters

Conversion of energy in motors

• Motors convert electrical energy into mechanical work through the interaction of magnetic fields and electric currents
  • Current-carrying loop experiences a torque when placed in an external magnetic field causing rotation and conversion of electrical energy to rotational kinetic energy
• Lorentz force $\vec{F} = I\vec{L} \times \vec{B}$ acts on current-carrying wires causing torque on the loop
  • $I$ represents current, $\vec{L}$ length vector of wire, and $\vec{B}$ magnetic field
  • Force is perpendicular to both current and magnetic field resulting in torque
• Torque direction depends on current direction and magnetic field orientation
  • Right-hand rule determines torque direction (fingers point in current direction, magnetic field lines into palm, thumb points in torque direction)
  • Fleming's left-hand rule can also be used to determine the direction of motion in motors

Torque calculation for current loops

• Torque on a current-carrying loop in a uniform magnetic field: $\vec{\tau} = \vec{\mu} \times \vec{B}$
  • $\vec{\tau}$ represents torque vector
  • $\vec{\mu}$ magnetic dipole moment of loop, $\vec{\mu} = NIA\hat{n}$ ($N$ number of turns, $I$ current, $A$ loop area, $\hat{n}$ unit vector normal to loop plane)
  • $\vec{B}$ external magnetic field
• Torque magnitude calculation: $\tau = \mu B \sin\theta$
  • $\theta$ angle between magnetic dipole moment and magnetic field
  • Maximum torque when loop perpendicular to magnetic field ($\theta = 90^\circ$), zero torque when loop parallel ($\theta = 0^\circ$ or $180^\circ$)
• Work done by motor related to torque and angular displacement
  • Work done: $W = \tau\Delta\theta$ ($\Delta\theta$ angular displacement)
• Angular momentum of the rotating loop contributes to the motor's inertia and affects its dynamic response

Function of brushes and commutators

• Brushes and commutators enable continuous rotation in DC motors
• Commutator is a split ring attached to and rotates with the armature (rotating part of motor)
  • Made of insulated conductive segments (usually copper)
  • Each segment connected to one end of armature windings
• Brushes are stationary contacts (carbon or graphite) pressing against the commutator
  • Connected to external circuit supplying current to motor
• As armature rotates, commutator segments contact brushes, reversing current direction in armature windings
  • Reversal ensures torque on armature always in the same direction for continuous rotation
  • Without commutator and brushes, armature would oscillate instead of rotating continuously
• Brushes and commutator also transfer electrical power from stationary part (stator) to rotating armature

Electromagnetic Principles in Motors

• Magnetic flux through the loop affects the motor's performance and efficiency
• Electromagnetic induction plays a crucial role in the operation of motors, generating back EMF as the loop rotates in the magnetic field
• Changes in magnetic flux induce currents in the loop, following Faraday's law of induction