23.1 Induced Emf and Magnetic Flux

Magnetic flux measures the total magnetic field passing through a surface. It's calculated using the equation Φ_B = BA cos θ, where B is the field strength, A is the surface area, and θ is the angle between them.

Faraday's law states that a changing magnetic flux induces an electromotive force (emf) in a loop. The induced emf is given by ε = -dΦ_B/dt, where dΦ_B/dt is the rate of change of magnetic flux.

Magnetic Flux and Induced EMF

Magnetic flux calculation

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• Magnetic flux ($\Phi_B$) measures the total magnetic field passing through a surface
  • Calculated using the equation: $\Phi_B = \vec{B} \cdot \vec{A} = BA\cos\theta$
    • $\vec{B}$: magnetic field strength (teslas, T)
    • $\vec{A}$: area of the surface (square meters, m²)
    • $\theta$: angle between the magnetic field lines and the normal to the surface
• Magnetic field perpendicular to the surface ($\theta = 0°$) results in maximum flux: $\Phi_B = BA$
  • Example: a flat coil placed perpendicular to a uniform magnetic field
• Magnetic field parallel to the surface ($\theta = 90°$) results in zero flux: $\Phi_B = 0$
  • Example: a flat coil placed parallel to a uniform magnetic field
• Flux linkage: the total magnetic flux passing through all turns of a coil

Induction of electromotive force

• Faraday's law of induction states a changing magnetic flux through a loop induces an electromotive force (emf) in the loop
  • Induced emf ($\mathcal{E}$) given by: $\mathcal{E} = -\frac{d\Phi_B}{dt}$
    • $\frac{d\Phi_B}{dt}$: rate of change of magnetic flux
    • Negative sign indicates the induced emf opposes the change in flux (Lenz's law)
• Changing magnetic flux caused by:
  • Changing the magnetic field strength (increasing or decreasing the field)
  • Changing the area of the loop (expanding or contracting the loop)
  • Changing the orientation of the loop relative to the magnetic field (rotating the loop)
• Motion of a conductor in a magnetic field also induces an emf
  • Motional emf: $\mathcal{E} = Blv$
    • $B$: magnetic field strength
    • $l$: length of the conductor
    • $v$: velocity of the conductor perpendicular to the magnetic field
  • Example: a conducting rod moving through a magnetic field

Factors affecting induced emf

• Magnitude of induced emf depends on:
  1. Rate of change of magnetic flux
     • Faster changes in flux result in larger induced emf
     • Example: rapidly moving a magnet in and out of a coil
  2. Number of turns in a coil (for coiled conductors)
     • More turns lead to a larger induced emf
     • Example: a tightly wound coil with many turns
• Direction of induced emf (and resulting current) determined by Lenz's law
  • Induced emf always opposes the change in magnetic flux that caused it
  • Applying the right-hand rule helps determine the direction of induced current
• Practical applications:
  • Generators convert mechanical energy into electrical energy using induced emf (hydroelectric, wind turbines)
  • Transformers use induced emf to change voltage levels in electrical systems (power grids)
  • Induction cooktops use induced currents to heat cookware (efficient, safe cooking)
  • Eddy current brakes use induced currents to slow down moving objects (roller coasters, trains)

Inductance and Magnetic Properties

• Mutual inductance: the property of two circuits where a change in current in one induces an emf in the other
• Self-inductance: the property of a circuit where a change in its own current induces an emf in itself
• Magnetic permeability: a measure of how easily a material can be magnetized in response to an applied magnetic field
• Magnetic dipole moment: a measure of the strength and orientation of a magnetic dipole, such as a current loop or a bar magnet