Faraday's Law and Magnetic Induction

Introduction to Magnetic Induction and Faraday's Law

• Electric circuits are essential in modern devices, requiring an electromotive force (EMF) for current flow.
• Electric generating stations produce EMF, converting energy such as gravitational potential, chemical, or nuclear energy.
• Electromagnetic induction describes when a changing magnetic flux induces EMF and current in a circuit.
• Transformers depend on magnetically induced EMFs.
• Faraday's law relates induced EMF to changing magnetic flux in any loop, including closed circuits.

Induction Experiments

• Michael Faraday and Joseph Henry conducted pioneering experiments on magnetically induced EMF in the 1830s.
• Key observations from experiments with coils and magnets include:
  • A stationary magnet near a coil shows no current.
  • Moving a magnet toward or away from the coil induces current.
  • Moving the coil instead of the magnet also induces current.
  • Replacing the magnet with a second coil connected to a battery induces current only when one coil is moving relative to the other.
  • Varying the current in one coil (by opening/closing the switch or changing resistance) induces current in a nearby stationary coil.
• Induced current refers to the current generated this way, and induced EMF is the electromotive force required to cause it.
• Changing magnetic flux is the common element in these experiments.
• Induced EMF is proportional to the rate of change of magnetic flux; its direction depends on whether the flux is increasing or decreasing.
• Induced EMFs have practical applications such as electric generators, and wall sockets.

Magnetic Flux

• Magnetic flux (Φ or ΦB) describes the component of the magnetic field passing through a surface.
• It's proportional to the number of field lines through the surface.
• The magnetic flux through a surface of vector area A is ΦB = B ⋅ A = BA cosθ, where:
  • B is the magnetic field magnitude in Tesla (T).
  • A is the area of the surface.
  • θ is the angle between magnetic field lines and the normal (perpendicular) to A.
• The SI unit for magnetic flux is the tesla-square meter (T⋅m²), also known as the weber (Wb) where 1 Wb = 1 T⋅m².

Faraday's Law

• It states that the induced EMF in a closed loop is equal to the negative of the time rate of change of magnetic flux through the loop.
• In symbols, Faraday's law is written as ε = -dΦB/dt.
• The induced EMF opposes the flux change.
• If the magnetic flux changes through a coil of N turns, the total induced EMF is ε = -N(dΦB/dt).
• Change the magnetic flux through a coil by:
  • Change the magnitude B of the magnetic field within the coil.
  • Alter the total area of the coil or the portion within the magnetic field.
  • Change the angle between the magnetic field direction and the plane of the coil.

Example: EMF and Current Induced in a Loop

• Consider a uniform magnetic field that is increasing at a rate of 0.020 T/s, passing through a conducting loop of 120 cm² area and 5.0 Ω resistance.
• To find the induced EMF and current, calculate the magnetic flux ΦB = BA, then use Faraday's law to find ε, and Ohm's law (ε = IR) to find I.
• Replacing a conducting loop with an insulator doesn't change the induced EMF.
• The current will be smaller; if the loop is a perfect insulator, the induced current is zero.

Direction of Induced EMF

• To find the direction of an induced EMF or current:
  • Define a positive direction for the vector area 𝐴.
  • Determine the sign of the magnetic flux ΦB and its rate of change based on the directions of 𝐴 and the magnetic field 𝐵.
  • Determine the sign of the induced EMF or current: If the flux is increasing, the induced EMF/current is negative; if decreasing, the induced EMF/current is positive.
  • Use the right-hand rule to find the direction of the induced EMF or current.

Lenz's Law

• Lenz's law states that any induction effect tends to oppose the change that caused it.
• It's a manifestation of the conservation of energy.
• Electromagnetic induction can be used to produce voltage using magnetic fields, generating current in a closed circuit.
• Factors determining the induced EMF:
  • Increase the number of wire turns in the coil to increase the induced EMF.
  • Increase the relative speed between the coil and magnet to increase the induced EMF.
  • Increase the strength of the magnetic field for greater EMF.

Applications of Faraday's Law

• Faraday's law has applications:
  • Electrical equipment like transformers.
  • Induction cookers.
  • Electromagnetic flowmeters.
  • Electric guitars and violins.
  • Maxwell's equation

Example: Magnitude and Direction of an Induced EMF

• A 500-loop coil with a 4.00 cm radius is placed in a magnetic field decreasing at 0.200 T/s:
  • Calculating the induced EMF using Faraday's law: ε = N(dB/dt)Acosθ.
  • The direction of the induced EMF follows the right-hand rule.

Induced Electric Field

• A changing magnetic flux results in an induced EMF.
• Faraday's law states that induced EMF is the negative rate of change of magnetic flux.
• If it changes over time, it has to be an induced EMF, then charges can move and produce a current.
• This current is referred to as an induced current.
• An induced electric field makes the charges move.
• This induced electric field is non-electrostatic, they go around in loops, appearing when a magnetic flux is present, and disappear when there is no change.
• The induced electric field is nonconservative.