7.1-7.2. Electromagnetic Induction

Ohm's Law

Electric current occurs when charge is in motion. Inside a substance, you have to keep pushing the charges to keep them moving. For most substances, the current density is proportional to the force per unit charge:

$\mathbf{J}=\sigma\mathbf{f}$

If we only consider electric force (assuming the electron moves not too fast, so magnetic force can be ignored), it becomes

$\mathbf{J}=\sigma\mathbf{E}$

This is called Ohm's Law. The proportionality factor $\sigma$ is called the conductivity of the medium. Its reciprocal, $\rho=1/\sigma$ , is called the resistivity of the medium. The more familiar version of this law,

$V=IR$

is obtained by integrating the field and current density over the wire from an electrode to the other.

One may question: if the charge is being applied force, why is it not accelerating, resulting in increasing current? It is because energy is dissipated due to frequent collisions of electrons inside the material. (largely due to thermal motion, in classical theory) The dissipated work is converted into heat. The amount of energy is described by the Joule heating law:

$P=VI=I^2R$

Electromotive Force

Think of a simple electric circuit, where the wire loop connects the battery and the resistor. The battery pushes the charges(electron) along the circuit. When an electron near the electrode is pushed, the neighboring electrons get affected by the electrostatic force. So, there is no local concentration of electrons compared to other sections, and a uniform current flows along the entire wire loop. The net effect of the source(in this case, battery) can be obtained by

$\displaystyle \varepsilon = \oint{\mathbf{f}\cdot d\mathbf{l}}$

This is called the electromotive force, or emf. Despite its name, the quantity is not a force. Instead, it can be interpreted as the work done for each unit charge. Many things can be the source of emf, including electrochemical batteries, mechanical force, or light in a photoelectric cell. A generator is also one of the sources. Motional emfs arise in them, when you move a wire through a magnetic field.

Motional emfs arise primarily because the magnetic field exerts Lorentz force to moving electrons in the wire. For a wire loop, the motional emf can be calculated through the following flux rule:

$\displaystyle \varepsilon = -{d\Phi\over dt}$

It means that the emf arises in a wire loop, when the magnetic flux passing the loop changes. Note that this rule is derived from Lorentz force law.

Faraday's law

A series of experiments conducted by Michael Faraday in 1831 revealed that moving a wire loop inside a magnetic field made a current flow in the wire loop. Faraday concluded that a changing magnetic field induces an electric field.

Recall the flux rule. When we plug in the B field, we get

$\displaystyle \oint{\mathbf{E}\cdot d{\mathbf{l}}}=-\int{{\partial\mathbf{B}\over\partial t}\cdot d\mathbf{a}}$

which is (the integral form of) Faraday's law. By applying Stoke's theorem, we get the differential form:

$\displaystyle \nabla \times \mathbf{E}=-{\partial\mathbf{B}\over \partial t}$

When we calculate the electric field using Faraday's law (integral form), we often get confused with the direction. Lenz's law states the direction of the induced electric field: The electric field is induced in a direction that cancels the change of magnetic flux. Nature abhors the change in flux.

In electrostatics, we have seen that curl of electric field is always zero. However, when magnetic field changes over time, the electric field becomes nonzero. In a pure Faraday field (where $\rho=0$ everywhere), we have

$\displaystyle \nabla\cdot\mathbf{E}=0,\quad\nabla\times\mathbf{E}=-{\partial\mathbf{B}\over dt}$

whose forms resemble to what we saw in magnetostatics.

$\displaystyle\nabla\cdot\mathbf{B}=0,\quad\nabla\times\mathbf{B}=-{\mu_0\mathbf{J}}$

Inductance

Consider two wire loops. when current flows along the first loop, the magnetic field would be induced. Then what is the magnetic flux passing the second loop? When we observe the Biot-Savart law, you can find out that the flux is proportional to the current in the first loop. That is,

$\Phi_2=M_{21}I_1$

for some proportionality factor $M_{21}$ . This is called the mutual inductance of the two loops. This quantity depends only on the geometry - sizes, shapes, and the distances - of the two loops. Then, we know another thing: mutual inductances are the same when current flows in loop 2 and we calculate the flux passing loop 1. $M_{12}=M_{21}$ .

When current flows in loop 1, the magnetic field also makes a flux inside the loop 1 itself. The flux is also proportional to the current. the proportionality factor $L$ here is called the self inductance.

$\Phi=LI$

The inductance is measured in Henries [H], or volt-second per ampere.

References

1. David J. Griffith. <Introduction to Electrodynamics>. 5th ed. Chapters 7.1-7.2