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Electrodynamics

Ohm's Law

The electric current through a conductor between two points is directly proportional to the voltage across the two points.

$V=IR\quad {\text{or}}\quad I={\frac {V}{R}}\quad {\text{or}}\quad R={\frac {V}{I}}$

The vector form of the law used in electromagnetics and material science is:

$\mathbf {J} =\sigma \mathbf {E} ,$ where J is the current density at a given location in a resistive material, E is the electric field at that location, and σ(sigma) is a material-dependent parameter called the conductivity, defined as the inverse of resistivity ρ (rho).

EMF (Electromotive force)

An energy transfer to an electric circuit per unit of electric charge, measured in volts.

Inside a source of emf (such as a battery) that is open-circuited, a charge separation occurs between the negative terminal N and the positive terminal P, An electrostatic field that points from P to N is formed, whereas the emf of the source must be able to drive current from N to P when connected to a circuit. This led Max Abraham to introduce the concept of a nonelectrostatic field

that exists only inside the source of emf. In the open-circuit case, ⁠⁠, while when the source is connected to a circuit the electric field inside the source changes but remains essentially the same. In the open-circuit case, the conservative electrostatic field created by separation of charge exactly cancels the forces producing the emf. Expressed mathematically:

${\mathcal {E}}_{\mathrm {source} }=\int _{N}^{P}{\boldsymbol {E}}'\cdot \mathop {} \!\mathrm {d} {\boldsymbol {\ell }}=-\int _{N}^{P}{\boldsymbol {E}}_{\mathrm {open\ circuit} }\cdot \mathop {} \!\mathrm {d} {\boldsymbol {\ell }}=V_{P}-V_{N}\ ,$

where is the conservative electrostatic field created by the charge separation associated with the emf, is an element of the path from terminal N to terminal P, ⋅ (dot operator) denotes the vector dot product, and is the electric scalar potential.This emf is the work done on a unit charge by the source's nonelectrostatic field when the charge moves from N to P.

Induction

Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The electric current produces a magnetic field around the conductor. The magnetic field strength depends on the magnitude of the electric current, and therefore follows any changes in the magnitude of the current. From Faraday's law of induction, any change in magnetic field through a circuit induces an electromotive force (EMF) (voltage) in the conductors, a process known as electromagnetic induction. This induced voltage created by the changing current has the effect of opposing the change in current. This is stated by Lenz's law, and the voltage is called back EMF.

Faraday's Law (of induction)

In electromagnetism, Faraday's law of induction describes how a changing magnetic field can induce an electric current in a circuit. This phenomenon, known as electromagnetic induction, is the fundamental operating principle of transformers, inductors, and many types of electric motors, generators and solenoids.

Alternating electric current flows through the solenoid on the left, producing a changing magnetic field. This field causes, by electromagnetic induction, an electric current to flow in the wire loop on the right.

Faraday's law of induction, also known as the flux rule, flux law, and Faraday–Lenz law,^[1] states that the electromotive force (emf) around a closed circuit is equal to the negative rate of change of the magnetic flux through the circuit. This rule holds for any circuit made of thin wire and accounts for changes in flux due to variations in the magnetic field, movement of the circuit, or deformation of its shape.^[2] The direction of the induced emf is given by Lenz's law, which states that the induced current will flow in such a way that its magnetic field opposes the change in the original magnetic flux.^[3]

Mathematically, in SI units, the law is expressed as ${\mathcal {E}}=-{\frac {\mathrm {d} \Phi _{B}}{\mathrm {d} t}},$ where ${\mathcal {E}}$ is the electromotive force (emf) and $Φ B$ is the magnetic flux through the circuit. The magnetic flux is defined as the surface integral of the magnetic field $B$ over a time-dependent surface $Σ(t)$ , whose boundary is the wire loop: $\Phi _{B}=\iint _{\Sigma (t)}\mathbf {B} (t)\cdot \mathrm {d} \mathbf {A} \,,$ where $d A$ is an infinitesimal area vector normal to the surface. The dot product $B \cdot d A$ represents the flux through the differential area element. In more visual terms, the magnetic flux is proportional to the number of magnetic field lines passing through the loop.

When the flux changes, an emf is induced around the loop. This emf corresponds to the energy per unit charge required to move it once around the loop.^[4]^[5]^[6] In a simple circuit with resistance $R$ , an emf ${\mathcal {E}}$ gives rise to a current $I$ according to the Ohm's law ${\mathcal {E}}=IR$ .^[7] Equivalently, if the loop is broken to form an open circuit and a voltmeter is connected across the terminals, the emf is equal to the voltage measured across the open ends.^[8]

For a tightly wound coil of wire, composed of $N$ identical turns, the same magnetic field lines cross the surface $N$ times. In this case, Faraday's law of induction states that^[9]^[10] ${\mathcal {E}}=-N{\frac {\mathrm {d} \Phi _{B}}{\mathrm {d} t}}$ where $N$ is the number of turns of wire and $Φ B$ is the magnetic flux through a single loop. The product $N Φ B$ is known as linked flux.^[11]

Week 2

Energy in magnetic field

Week 3

Displacement current

Maxwell’s equations

Wave equation

Week 4

Plane waves in vacuum

Polarization

Poynting vector and field momentum

Week 5

Maxwell equations and waves in matter

General boundary conditions

Boundary conditions in linear media

Week 6

Reflection and transmission

Snell’s law. Total internal reflection

Week 7

Fresnel equations.

Brewster’s angle

Waves in conductors.

Skin effect

Week 8

Drude model

Group velocity

Complex susceptibility and conductivity

Oscillator strengths

Week 9

Kramers-Kronig relations

Negative refraction index materials

Week 10

Propagation between plates

waveguides

Potentials

Gauge transformations

Week 11

Retarded potentials

Electric and magnetic dipole radiation

Week 12

Radiation from a localized distribution

(Electromagnetic radiation)

Liénard-Wiechert potentials

Overview of radiation by a point charge

Week 13

Lorentz transformations

Minkowski space

4-vectors of kinematics & dynamics

Week 14

Force of a line current on a moving charge

4-vectors of E&M: current density & potential

Field tensor and field transformations

(Special relativity)

Week 15

Field of a uniformly moving charge

Cherenkov radiation.

Radiation reaction

↑ Fujimoto, Minoru (2007-09-06). Physics of Classical Electromagnetism. New York: Springer Science & Business Media. p. 105. ISBN 978-0-387-68018-7.
↑ Landau, Lev Davidovich; Lifshitz, Evgeniĭ Mikhaĭlovich; Pitaevskiĭ, Lev Petrovich (1984). Electrodynamics of Continuous Media. Oxford: Pergamon press. p. 219. ISBN 0-08-030276-9.
↑ Griffiths 2023, p. 319.
↑ Feynman, Leighton & Sands 2006, Ch. 17.
↑ Griffiths 2023, pp. 304–306.
↑ Tipler; Mosca (2004). Physics for Scientists and Engineers. Macmillan. p. 795. ISBN 9780716708100.
↑ Zangwill 2013, pp. 462–464.
↑ Paul, Clayton R.; Scully, Robert C.; Steffka, Mark A. (2022-11-01). Introduction to Electromagnetic Compatibility. John Wiley & Sons. p. 703. ISBN 978-1-119-40434-7.
↑ Whelan, P. M.; Hodgeson, M. J. (1978). Essential Principles of Physics (2nd ed.). John Murray. ISBN 0-7195-3382-1.
↑ Nave, Carl R. "Faraday's Law". HyperPhysics. Georgia State University. Retrieved 2011-08-29.
↑ "121-11-77: "linked flux"". IEC 60050 - International Electrotechnical Vocabulary. Retrieved 2025-06-20.

[r127-1] Fujimoto, Minoru (2007-09-06). Physics of Classical Electromagnetism. New York: Springer Science & Business Media. p. 105. ISBN 978-0-387-68018-7.

[n495-2] Landau, Lev Davidovich; Lifshitz, Evgeniĭ Mikhaĭlovich; Pitaevskiĭ, Lev Petrovich (1984). Electrodynamics of Continuous Media. Oxford: Pergamon press. p. 219. ISBN 0-08-030276-9.

[FOOTNOTEGriffiths2023319-3] Griffiths 2023, p. 319.

[FOOTNOTEFeynmanLeightonSands2006Ch._17-4] Feynman, Leighton & Sands 2006, Ch. 17.

[FOOTNOTEGriffiths2023304–306-5] Griffiths 2023, pp. 304–306.

[6] Tipler; Mosca (2004). Physics for Scientists and Engineers. Macmillan. p. 795. ISBN 9780716708100.

[FOOTNOTEZangwill2013462–464-7] Zangwill 2013, pp. 462–464.

[f437-8] Paul, Clayton R.; Scully, Robert C.; Steffka, Mark A. (2022-11-01). Introduction to Electromagnetic Compatibility. John Wiley & Sons. p. 703. ISBN 978-1-119-40434-7.

[9] Whelan, P. M.; Hodgeson, M. J. (1978). Essential Principles of Physics (2nd ed.). John Murray. ISBN 0-7195-3382-1.

[10] Nave, Carl R. "Faraday's Law". HyperPhysics. Georgia State University. Retrieved 2011-08-29.

[11] "121-11-77: "linked flux"". IEC 60050 - International Electrotechnical Vocabulary. Retrieved 2025-06-20.

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