10.3 Motional emf and induced electric fields

Motional emf and induced electric fields are key concepts in electromagnetic induction. They explain how changing magnetic fields create electric fields and voltages in conductors, forming the basis for generators and transformers.

These phenomena demonstrate the interplay between electricity and magnetism. Understanding them is crucial for grasping how energy can be converted between different forms, a fundamental principle in many modern technologies.

Motional emf and Lorentz Force

Generating Motional emf

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• Motional emf is the electromotive force (voltage) induced in a conductor moving through a magnetic field
  • Occurs due to the magnetic force acting on the charges within the conductor
  • Magnitude depends on the velocity of the conductor, strength of the magnetic field, and length of the conductor
  • Direction determined by the right-hand rule (thumb points in the direction of motion, fingers in the direction of the magnetic field, and palm facing the direction of the induced current)
• Formula for motional emf: $\mathcal{E} = vBL\sin\theta$
  • $v$ is the velocity of the conductor
  • $B$ is the strength of the magnetic field
  • $L$ is the length of the conductor
  • $\theta$ is the angle between the velocity and magnetic field vectors
• Examples:
  • A metal rod moving through a uniform magnetic field (perpendicular to the field) will have an induced emf across its ends
  • A spinning disk in a magnetic field (Faraday disk) generates an emf between the center and the rim

Lorentz Force and Its Applications

• Lorentz force is the force experienced by a moving charged particle in the presence of an electromagnetic field
  • Combines the effects of both electric and magnetic fields on the particle
  • Formula for Lorentz force: $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$
    • $q$ is the charge of the particle
    • $\vec{E}$ is the electric field
    • $\vec{v}$ is the velocity of the particle
    • $\vec{B}$ is the magnetic field
• Lorentz force is responsible for the deflection of charged particles in magnetic fields
  • Used in applications such as mass spectrometers, particle accelerators, and cathode ray tubes (CRTs)
• Examples:
  • Electrons in a CRT television are deflected by magnetic fields to create images on the screen
  • Charged particles in the Van Allen radiation belts around Earth are trapped by the planet's magnetic field due to the Lorentz force

Conducting Rod in a Magnetic Field

• When a conducting rod moves through a magnetic field, an emf is induced along its length
  • Free electrons in the rod experience a Lorentz force, causing them to move and create a current
  • The induced emf opposes the change in magnetic flux (Lenz's law)
• The induced current in the rod can be used to power an external circuit
  • Sliding contacts at the ends of the rod allow the current to flow through the external circuit
• Examples:
  • A sliding conducting rod between two parallel rails in a magnetic field can be used as a simple generator
  • The Faraday disk (a spinning metal disk in a magnetic field) is another example of a conducting rod in a magnetic field, where the rod is essentially a radial segment of the disk

Induced Electric Fields

Properties of Induced Electric Fields

• An induced electric field is created by a changing magnetic field, as described by Faraday's law
  • The changing magnetic flux through a surface generates an electromotive force (emf) that drives the induced electric field
• Induced electric fields are non-conservative, meaning the work done by the field on a charged particle moving in a closed loop is non-zero
  • This is in contrast to electrostatic fields, which are conservative
• The direction of the induced electric field is determined by Lenz's law
  • The induced field opposes the change in magnetic flux that created it
• Examples:
  • A changing magnetic field through a loop of wire induces an electric field in the wire, causing a current to flow
  • Electromagnetic induction in transformers relies on induced electric fields in the secondary coil due to the changing magnetic field from the primary coil

Non-Conservative Nature of Induced Electric Fields

• Non-conservative fields are characterized by the presence of a non-zero curl ($\nabla \times \vec{E} \neq 0$)
  • The curl of an induced electric field is proportional to the rate of change of the magnetic field ($\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$)
• The work done by a non-conservative field on a charged particle moving in a closed loop is non-zero
  • This is because the field is not the gradient of a scalar potential ($\vec{E} \neq -\nabla V$)
• The presence of an induced electric field allows for the transfer of energy between magnetic fields and electric currents
  • This is the basis for many applications, such as generators and transformers
• Examples:
  • In a transformer, the non-conservative nature of the induced electric field in the secondary coil allows for the transfer of energy from the primary coil
  • In a generator, the non-conservative induced electric field drives the current in the armature, converting mechanical energy into electrical energy

Faraday's Law in Terms of Electric Field

• Faraday's law can be expressed in terms of the induced electric field:
  • $\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$
    • The line integral of the induced electric field around a closed loop equals the negative rate of change of the magnetic flux through the surface bounded by the loop
• This form of Faraday's law relates the spatial properties of the induced electric field to the temporal changes in the magnetic flux
• The minus sign in the equation represents Lenz's law, indicating that the induced electric field opposes the change in magnetic flux
• Examples:
  • In a solenoid, the changing current in the windings creates a changing magnetic flux, which induces an electric field along the length of the solenoid
  • In a moving conductor, the changing magnetic flux due to the motion of the conductor induces an electric field within the conductor, leading to the phenomenon of motional emf