Electromagnetic induction is the phenomenon in which a changing magnetic flux produces an electric current in a circuit. It was discovered independently by Michael Faraday in 1831 and Joseph Henry. Faraday's discovery showed that magnetism could be converted into electricity, and it is the principle behind generators, transformers, and many electrical devices.
Faraday quantified the phenomenon in his law of induction, and Lenz's law gives the direction of the induced current. The chapter develops the concept of magnetic flux, the induced EMF, and motional EMF, where a conductor moving in a magnetic field generates a voltage. We also study eddy currents and their uses.
The second half of the chapter introduces inductance. Self-inductance relates the flux linked with a circuit to the current through it, storing energy in the magnetic field as U = (1/2)LI^2. Mutual inductance relates the flux in one coil to the current in another, and is the basis of the transformer. The AC generator converts mechanical energy into electrical energy by this very principle.
Magnetic flux through a surface is the product of the magnetic field and the area perpendicular to it. For a uniform field B through a plane area A, whose normal makes angle theta with the field:
Phi = B A cos(theta) = B dot A
The SI unit of magnetic flux is the weber (Wb), where 1 Wb = 1 T m^2. Flux is a scalar quantity. The flux is maximum when the field is perpendicular to the area (theta = 0) and zero when the field is parallel to the surface (theta = 90 degrees).
The flux linked with a coil of N turns is N times the flux through a single turn. Faraday's law depends on the rate of change of this flux, whether it changes because the field changes, the area changes, or the orientation changes.
Faraday's law of electromagnetic induction states that the magnitude of the induced EMF in a circuit is equal to the rate of change of magnetic flux through it:
E = - d Phi / d t
For a coil of N turns, the induced EMF is:
E = - N d Phi / d t
The negative sign embodies Lenz's law, which states that the induced current flows in a direction that opposes the change in flux that produces it. Lenz's law is a consequence of the conservation of energy: the induced current's magnetic field tries to maintain the original flux.
To apply Lenz's law, find the direction of the change in flux, then determine the direction of the induced current whose field opposes that change. If the flux is increasing, the induced current produces a field opposing it; if decreasing, the induced current supports it. Lenz's law determines the direction, while Faraday's law gives the magnitude.
When a conductor of length l moves with velocity v perpendicular to a uniform magnetic field B, the charges inside experience a magnetic force that separates them, creating a potential difference. This induced EMF is called the motional EMF:
E = B l v
The motional EMF is maximum when the velocity is perpendicular to both the field and the conductor. If the velocity makes an angle theta with the field, the EMF is E = B l v sin(theta). The direction is given by Lenz's law or Fleming's right-hand rule: the forefinger along the field, the thumb along the motion, and the middle finger gives the direction of the induced current.
A conducting rod sliding on rails in a magnetic field experiences a magnetic force that opposes its motion, consistent with Lenz's law and energy conservation. The work done against this force appears as the electrical energy in the circuit.
When the magnetic flux linked with a solid conductor changes, induced currents circulate within the conductor in closed paths. These are called eddy currents. They were discovered by Foucault and are also known as Foucault currents.
Eddy currents are sometimes undesirable because they produce heating and energy loss. This is minimized by laminating the core of transformers and motors - building the core from thin insulated sheets, which restricts the eddy currents to small regions. The resistance of each lamina is high, reducing the induced currents.
Eddy currents are also useful. They are used in electromagnetic braking in trains and cranes, in induction furnaces for melting metals, and in speedometers. Eddy currents always oppose the change producing them, in accordance with Lenz's law, which is why electromagnetic brakes work smoothly.
Self-inductance is the property of a coil by which it opposes a change in the current through itself. When the current changes, the flux linked with the coil changes, inducing an EMF in the coil itself:
E = - L d I / d t
The self-inductance L of a solenoid with n turns per unit length, length l, and area A is:
L = mu_0 n^2 A l = mu_0 N^2 A / l
The unit of inductance is the henry (H), where 1 H = 1 Wb/A = 1 V s/A. The induced EMF opposes the change in current; hence the name self-inductance. Because of this, inductors resist sudden changes in current.
The energy stored in the magnetic field of an inductor is:
U = (1/2) L I^2
This energy is stored in the magnetic field, analogous to the energy stored in the electric field of a capacitor. The magnetic energy density is u = B^2/(2 mu_0).
Mutual inductance is the property by which a change in current in one coil induces an EMF in a neighbouring coil. If current I1 in coil 1 produces a flux N2 Phi2 through coil 2, the mutual inductance is:
M = N2 Phi2 / I1
The induced EMF in coil 2 is:
E2 = - M d I1 / d t
The mutual inductance depends on the geometry of the coils, the number of turns, and the relative position and orientation. It is maximum when the coils are close and their axes aligned, and it is the same for both coils: M12 = M21. Mutual inductance is the basis of the transformer.
The AC generator converts mechanical energy into electrical energy by electromagnetic induction. It consists of a coil of N turns and area A rotating in a uniform magnetic field B with angular speed omega. The flux through the coil at time t, when its normal makes angle omega t with the field, is:
Phi = B A cos(omega t)
The induced EMF is:
E = N B A omega sin(omega t)
The EMF varies sinusoidally with time, with peak value E0 = N B A omega. The frequency of the alternating EMF equals the rotation frequency of the coil. The generator thus produces alternating current, the standard form of electrical power supply. The alternating current produced by generators is converted, transmitted, and used in homes and industries.
| Quantity | Formula | Unit |
|---|---|---|
| Magnetic flux | Phi = B A cos theta | weber (Wb) |
| Faraday's law | E = -d Phi/dt | volt (V) |
| Coil of N turns | E = -N d Phi/dt | volt (V) |
| Motional EMF | E = B l v | volt (V) |
| Self-induced EMF | E = -L dI/dt | volt (V) |
| Self-inductance of solenoid | L = mu_0 n^2 A l | henry (H) |
| Energy in inductor | U = (1/2) L I^2 | joule (J) |
| Mutual induced EMF | E2 = -M dI1/dt | volt (V) |
| Generator EMF | E = N B A omega sin(omega t) | volt (V) |
| Magnetic energy density | u = B^2/(2 mu_0) | J/m^3 |
| Application | Principle |
|---|---|
| Transformer | Mutual inductance |
| Generator | E = -d Phi/dt |
| Electromagnetic brake | Eddy currents oppose motion |
| Laminated cores | Reduce eddy current loss |
| Induction furnace | Eddy current heating |
This chapter established the link between changing magnetic flux and induced EMF. Faraday's law E = -N dPhi/dt and Lenz's law together give both the magnitude and direction of induction. We derived the motional EMF E = Blv for a moving conductor and studied eddy currents, which can cause losses but also enable braking and induction heating. Self-inductance with energy (1/2)LI^2 and mutual inductance M (the basis of the transformer) completed the circuit of ideas. Finally, the AC generator, producing E = NBA omega sin(omega t), showed how mechanical energy is converted into the alternating current that powers the modern world.