Premed · Premed · Physics 2
Lecture 11: Electromagnetic Induction and Faraday's Law
Physics II — Electromagnetism, Optics & Modern Physics
Learning Objectives
By the end of this lecture, students will be able to:
- Describe electromagnetic induction and the conditions required to generate an induced EMF
- State and apply Faraday's law of induction to calculate induced EMF
- Apply Lenz's law to determine the direction of induced current
- Calculate the EMF induced in a moving conductor (motional EMF)
- Explain the principles behind generators, transformers, and eddy currents
Lecture Content
I. Electromagnetic Induction: Discovery and Concepts
Michael Faraday in 1831 and Joseph Henry independently discovered electromagnetic induction, one of the most important phenomena in all of physics. The key observation is that a changing magnetic field induces an electric current in a nearby conductor. Induction occurs under a variety of circumstances: when a magnet moves relative to a coil, when the current in a nearby coil changes, when a coil moves into or out of a magnetic field, when the area of a loop in a magnetic field changes, or when the orientation of a loop in a magnetic field changes. The common thread uniting all of these situations is that the magnetic flux through the loop must be changing.
II. Magnetic Flux
Magnetic flux (Phi_B) through a surface is defined as Phi_B = integral of B dot dA. For a uniform field passing through a flat surface, this simplifies to Phi_B = BA cos(theta), where theta is the angle between B and the area normal vector. The SI unit of magnetic flux is the weber (Wb) = 1 T m^2.
The flux can change due to changes in the magnitude of B, the area A of the loop, the angle theta between B and the surface normal, or any combination of these. Each of these changes can produce an induced EMF.
III. Faraday's Law of Induction
Faraday's law states that the induced EMF in a loop is equal to the negative rate of change of magnetic flux through the loop: EMF = -d(Phi_B)/dt. For a coil with N turns, the induced EMF is EMF = -N d(Phi_B)/dt. The magnitude of the induced EMF depends on how rapidly the flux changes, so faster changes produce larger EMFs.
The negative sign in Faraday's law embodies Lenz's law, which specifies the direction of the induced current. Faraday's law is the third of Maxwell's four equations in integral form and stands as one of the foundational laws of electromagnetism. If the circuit is closed with total resistance R, the induced EMF drives a current I_induced = EMF / R.
IV. Lenz's Law
Lenz's law states that the direction of the induced current is such that it opposes the change in magnetic flux that produced it. This is a direct consequence of conservation of energy: if the induced current reinforced the change rather than opposing it, we would get runaway energy creation, which is impossible.
To apply Lenz's law in practice, first determine the direction of the original magnetic flux through the loop. Then determine whether the flux is increasing or decreasing. The induced current will create a magnetic field that opposes the change: if the flux is increasing, the induced field opposes the external field (points in the opposite direction), and if the flux is decreasing, the induced field supports the external field (points in the same direction). Finally, use the right-hand rule to find the direction of the induced current from the direction of the induced field.
<image>Two panels illustrating Lenz's law. Panel A: A bar magnet with its north pole approaching a conducting loop from the left. The external flux through the loop is increasing (more field lines through the loop). The induced current flows counterclockwise (as viewed from the magnet) to create a magnetic field opposing the increasing flux — the loop acts like a magnet with its north pole facing the approaching magnet (repulsion). Panel B: The bar magnet is being pulled away from the loop. The flux is decreasing. The induced current flows clockwise to create a field that tries to maintain the flux — the loop acts like a magnet with its north pole facing the retreating magnet (attraction). Arrows indicate current direction, induced B field, and force on the magnet in both cases.</image>
V. Motional EMF
An EMF is induced in any conductor moving through a magnetic field. For a straight conductor of length L moving with velocity v perpendicular to a uniform field B, the induced EMF is EMF = BLv. The magnetic force on the charges in the wire (F = qv x B) drives them along the conductor, causing one end to become positive and the other negative, thereby creating a potential difference.
The classic sliding rail problem illustrates this beautifully. A conducting rod slides along parallel rails in a magnetic field, generating an EMF of BLv. The resulting current is I = BLv/R, where R is the circuit resistance. The current-carrying rod in the magnetic field then experiences a retarding force F = BIL = B^2 L^2 v / R, which opposes the rod's motion in accordance with Lenz's law. The power dissipated in the resistor is P = Fv = B^2 L^2 v^2 / R, which equals I^2 R, confirming energy conservation. The energy dissipated as heat comes from the work done by the external force pushing the rod.
<image>A diagram of the sliding rail problem. Two horizontal parallel conducting rails are connected by a resistor R on the left. A conducting rod slides to the right along the rails with velocity v. A uniform magnetic field B points into the page (x symbols). The rod has length L (the distance between the rails). The induced EMF = BLv drives a counterclockwise current I = BLv/R through the circuit. The magnetic force on the current-carrying rod (F = BIL) is shown pointing to the left, opposing the rod's rightward motion. The increase in enclosed area dA = L dx is shaded.</image>
VI. Applications of Electromagnetic Induction
Electric generators work by rotating a coil in a magnetic field, producing a sinusoidal EMF given by EMF = NAB omega sin(omega t), where omega is the angular frequency of rotation. The maximum EMF is EMF_max = NAB omega. This is the fundamental principle behind AC power generation.
Transformers consist of two coils (primary and secondary) wound on a shared iron core. A changing current in the primary coil creates a changing flux, which induces an EMF in the secondary coil. The voltage ratio is V_s/V_p = N_s/N_p (the turns ratio), and for an ideal transformer, power is conserved: V_p I_p = V_s I_s. A step-up transformer (N_s > N_p) increases voltage while decreasing current, and a step-down transformer (N_s < N_p) does the reverse. Transformers are essential for efficient long-distance power transmission, allowing electricity to be transmitted at high voltage (low current) to minimize I^2R losses in the transmission lines.
Eddy currents are induced currents that form closed loops within bulk conductors exposed to changing magnetic fields. They produce both heating effects, exploited in induction cooktops and induction furnaces, and braking effects, used in electromagnetic brakes on trains and roller coasters. In transformers, eddy currents are undesirable because they waste energy as heat; they are minimized by using laminated cores made of thin insulated sheets.
<image>A cutaway diagram of a transformer. An iron core (rectangular loop shape) has a primary coil with N_p turns wound on the left side and a secondary coil with N_s turns wound on the right side. Magnetic flux lines are shown circulating through the core. The primary is connected to an AC voltage source V_p, and the secondary is connected to a load resistance. The equations V_s/V_p = N_s/N_p and V_p I_p = V_s I_s are displayed. Labels indicate this is a step-up transformer (N_s > N_p).</image>
