The course (80 hours of lectures and 50 hours of classroom exercises) provides the student with an introduction to classical electromagnetism, starting from phenomenology and achieving an interpretation of the phenomena within a classical field theory, by introducing Maxwell equations.
Expected learning outcomes
At the end of the course the student 1.Will have learnt the fundamental laws of electrostatics and the formulation of the associated general problem in empty space and conducting media via the Poisson and Laplace equations; 2.Will have learnt how to formulate electrostatics in the presence of dielectric media; 3.Will have learnt the fundamental laws of magnetostatics; 4.Will have studied the main phenomena associated with electromagnetic induction and understood how to interpret them using the Faraday-Neumann-Lenz law; 5.Will have understood how to formulate electromagnetism through the Maxwell equations, by including the displacement current; 6.Will have derived the electromagnetic wave equation and understood the main properties of wave propagation; 7.Will have acquired the knowledge of the main properties of electromagnetic
Lesson period: year
(In case of multiple editions, please check the period, as it may vary)
Electrostatics Coulomb's Law. The Electric Field. Continuous Charge Distributions. Field Lines, Flux, and Gauss's Law. The Divergence of the electric field. Applications of Gauss's Law. The circulation of the electric field. The electric Potential. The Potential of a Localized Charge Distribution. Work and Energy in Electrostatics. The Energy of a system of point charges. The Energy of a continuous charge distribution. The energy of the electric field. Conductors: basic Properties. Conductors in electrostatic field. Induced charges. Surface charge. Poisson's Equation and Laplace's Equation. Solutions of Laplace's equation. Harmonic functions. Electrostatic Boundary Conditions and uniqueness theorems. Systems of curvilinear coordinates. Method of separation of Variables in Cartesian coordinates and in spherical coordinates. Solutions of Poisson's equations; method of images. Capacitance of a conductor. Capacitors. Energy stored in capacitors. Forces among capacitors plates. Systems of several conductors. Capacitance and potential coefficients. Electric dipoles. Approximate potential at large distances. Forces and torques on dipoles. Multipole expansion of the potential.
Electric Fields in Matter Dielectrics. Induced dipoles. Alignment of polar molecules Polarization. Linear dielectrics. Susceptibility, Permittivity, Dielectric Constant. Polarization charges. Physical interpretation of polarization charges. The electric field of a polarized object. Gauss's Law in the presence of dielectrics. The electric displacement D. Electrostatic problem in presence of dielectric. Boundary conditions. Boundary value problems with linear dielectrics. Energy in systems with dielectrics.
Electric currents Electric current and current density. Charge conservation and continuity equation. Steady currents. Electrical conductivity and Ohm's Law. Resistivity. Resistance and resistors. Classical model of electrical conductivity. Cross section for rigid spheres collision. Drift velocity. Mobility. Conductivity. Conductors, semiconductors, insulators. Energy dissipation in current flow. Joule effect. Electromotive force and voltaic cell. Circuits and circuit elements. Networks with voltage sources. Kirchhoff's laws. Current sources. Ideal voltage and current sources. Real voltage and current sources. Internal resistance. Slowly varying electrical currents. Charge and discharge of capacitors.
Magnetostatics Curl of a vector field. Stokes' theorem. Magnetic forces. Oersted experiment. The Lorentz force law. Magnetic fields. Properties of magnetic forces. The Biot-Savart Law. The Magnetic Field of a Steady Current. The Divergence of B. Nonexistence of magnetic charge. Curl of B. Sources of the magnetic field. Ampere's Law. Applications of Ampere's Law. Volume and surface current densities. Magnetic field of a current circular loop. Magnetic Vector Potential. Helmoltz's theorem. Examples of vector potential. Vector potential of a circular loop at large distance. Magnetic dipole. Magnetic field of a magnetic dipole. Forces and torques on magnetic dipoles.
Einstein's special theory of relativity Einstein's special theory of relativity postulates. Relativity of simultaneity. Lorentz length contraction and time dilatation. Lorentz transformations. Four-vectors. Lorentz transformations in four-dimensional formalism. Energy-Momentum four-vector. Relativistic invariance of electric charge. Electric field measured in different inertial frames of reference. Electric field of a point charge moving with constant velocity. Field of a point charge that starts or stops. Relativistic interpretation of magnetic force. Magnetic field measured in different inertial frames of reference. Electric and magnetic field Lorentz transformations in four-dimensional formalism.
Time varying electric and magnetic fields Electromotive force. Electromagnetic induction. Faraday's Law. Applications of Faraday's Law. Induction and motional electromotive force. Lenz's Law. The Induced Electric Field. Faraday's Law and Maxwell's equations. Mutual inductance and self-inductance. Inductors. Circuits with inductors. LR circuit. Magnetic energy. The LC oscillator. Electrodynamics: the displacement current and Maxwell's Equation's in vacuum.
Magnetic Fields in Matter How various substances respond to a magnetic field. Diamagnetic, paramagnetic, ferromagnetic substances. Atomic magnetic dipoles. Electron spin and magnetic moment. Magnetization and magnetic susceptibility. Microscopic theory of diamagnetism and paramagnetism. The magnetic field of a magnetized object. Magnetization volume and surface current densities. The magnetic intensity H. Ampère's Law in magnetized materials. Maxwell's equations in matter. Boundary conditions. Qualitative theory of ferromagnetism. Magnets. Linear and nonlinear media. Solution of magnetostatic problems with magnetized materials. Scalar magnetic potential.
Electrodynamics and electromagnetic waves. Electromagnetic waves. Wave equation for electric and magnetic field. Solutions of wave equation. Monochromatic plane waves. Polarization. Energy and momentum of electromagnetic field. Poynting's theorem. Momentum of the electromagnetic field. Maxwell's stress tensor. Energy and momentum of electromagnetic waves. Radiation pressure. Electromagnetic waves propagation in linear media. Reflection and transmission at normal and oblique incidence. Fundamental laws of geometrical optics. Potential formulation of electrodynamics. Gauge transformations and gauge invariance. Retarded potentials. Quasi-static approximation. Moving point charge: Lienard-Wiechert potentials. Radiation. Electric dipole radiation at large distance. Radiation from point charges. Energy radiated and Larmor's formula. Linear and circular accelerators. Hydrogen atom instability in classical electrodynamics. Covariant formulation of electrodynamics. Introduction to classical electron theory: electromagnetic mass and momentum.
Prerequisites for admission
Integral and differential calculus of real functions. Vector in R3 space. Basic operations on vectors. Basic concepts of mechanics as: forces, conservative forces, work, kinetic energy, potential energy. Newton Laws and differential equation of motion.
Classroom lectures (80 hours) Classroom problem solving (50 hours)
Suggested textbook (in english) David Griffiths Introduction to electrodynamics, fourth edition Cambridge; 4th ed. (2017)
Possible alternative textbook (in italian) P. Mazzoldi - M. Nigro - C. Voci Fisica Volume II seconda edizione - EdiSES (2008)
Edward M. Purcell, David J. Morin Electricity and Magnetism, third edition Cambridge University Press (2013)
Richard P. Feynman The Feynman Lectures on Physics, Vol. II various editions; more recent: Basic Books; New Millennium edition (2011) free access online edition http://www.feynmanlectures.caltech.edu/
Assessment methods and Criteria
Assessment methods: written and oral exam. Written exam consists of three exercises. Students can use up to three hours to complete the written test. Ability to perform numerical calculations is required. The oral exam (lasting from 45 to 60 minutes) aims to evaluate the body of knowledge achieved by the student, his/her ability to critically discuss electromagnetism laws as well as his/her ability to perform calculations rigorously.
1. Electrostatics Coulomb's law. The electric field. Continuous charge distributions. Field lines, flow, Gauss's law. The divergence of the electric field. Applications of Gauss's law. The electric field circulation. Curl of a vector field. Stokes' theorem. The electric potential. Examples. Work and energy in electrostatics. The energy of a point charge system. The energy of a continuous distribution of charges. Electric field energy. Curvilinear coordinate systems. Multipole expansion of potential. The electric dipole. Potential in the approximation of large distance. Forces and moments acting on the dipoles.
2. Electrostatics in conductors Conductors in electrostatic fields. Induced charges. Surface charge density. Capacity of a conductor. Conductor systems. Capacity and potential coefficients. Capacitors. Energy stored in a capacitor. Forces between the plates of a capacitor. Poisson equation and Laplace equation. Solutions of the Laplace equation. Harmonic functions. Boundary conditions in electrostatics and uniqueness theorems. Solutions of the Poisson equation. Image charge method.
3. Electric fields in matter Dielectrics. Induced dipoles. Polarization by deformation and orientation. Linear dielectrics. Susceptibility, permittivity, dielectric constant. Polarization charges. The electric field of bulk polarized matter. Gauss's law in the presence of dielectrics. The electric displacement D. Electrostatic problem in the presence of dielectrics. Boundary conditions. Formulation of boundary problems with linear dielectrics. Energy in systems with dielectrics.
4. Electric currents Electric current and current density. Charge conservation and continuity equation. Stationary currents. Electrical conductivity and Ohm's law. Resistivity. Resistance and resistors. Classic model of conductivity. Drift velocity. Conductors, semiconductors, insulators. Dissipation of energy in current conduction. Joule effect. Electromotive force. Circuits and circuit elements. Networks with voltage generators. Kirchhoff's laws. Ideal voltage and current generators. Real current and voltage generators. Internal resistance. Charge and discharge of a capacitor.
5. Magnetostatics Lorentz force. The magnetic induction vector B. Properties of magnetic forces. Motion of a charged particle in a magnetic field. Applications: mass spectrometry, cyclotrons. Mechanical actions on circuits. Laplace formulas. The magnetic field of a stationary current. Examples: wire, coil and solenoid magnetic field. The divergence of B. Non-existence of magnetic charges. Curl of B. Sources of the magnetic field. Ampère's law. Applications of Ampère's law. Volume and surface current density. Potential vector. Potential vector of a circular loop at a great distance. Magnetic dipole. Magnetic field of a dipole. Forces and moments of forces on magnetic dipoles.
6. Magnetic fields in matter Response of different types of materials to the magnetic field. Diamagnetic, paramagnetic, ferromagnetic materials. Atomic magnetic dipoles. Intrinsic angular momentum of the electron (spin) and magnetic moments. Magnetization and magnetic susceptibility. Microscopic theory of diamagnetism and paramagnetism. The magnetic field of a magnetized body. Current density of volume and surface magnetization. The magnetic field H. Ampère's law in magnetized materials. Maxwell's equations in matter. Boundary conditions. Qualitative theory of ferromagnetism. Hysteresis cycle. Magnets. Linear and nonlinear materials. Solving magnetostatic problems with magnetized materials.
7. Time-dependent electric and magnetic fields Electromotive force. Electromagnetic induction. Faraday's law. Applications of Faraday's law. The induced electric field. Faraday's law and Maxwell's equations. Mutual inductance and self-inductance. Inductors. Circuits with inductors. LR Circuit. Magnetic energy. Electrodynamics: displacement current and Maxwell's equations.
8. Electrodynamics and electromagnetic waves Electromagnetic waves. Wave equation for the electric field and the magnetic field. Wave equation solutions. Plane monochromatic waves. Polarization. Energy and momentum of the electromagnetic field. Poynting theorem. Momentum of the electromagnetic field. Electromagnetic wave energy and momentum. Radiation pressure. Propagation of electromagnetic waves in linear media. Waves in conductors. Skin effect. Formulation of electrodynamics through potentials. Gauge transformations and gauge invariance. Retarded potentials. Quasi-static approximation. Point charges in accelerated motion and their radiation. Radiation of the oscillating dipole at large distance.
9. Optics Reflection and transmission in cases of normal and oblique incidence. Fundamental laws of geometric optics. Polarization of reflected and refracted waves. Dispersive phenomena. Lorentz oscillator model. Refractive index: real and imaginary part. Resonance absorption. Clausius-Mossotti's law.
10. Special relativity, quadrivectors and covariant formulation of electrodynamics.
Prerequisites for admission
Vector calculus, knowledge of differential and integral calculus as provided by the Courses of Analysis I and II. Fundamental concepts of mechanics : dynamics laws, conservative forces, potential energy, momentum and angular momentum.
The course consists in classroom lectures and exercise sessions, in a traditional style. Exercises play a key role, given the objectives of the course and the exam. Even during theory lessons, examples and exercises are worked out. Moreover, mathematical tools that are needed are reminded.
The exam consists in a written and oral part. The written exam lasts three hours and consists in solving typically three exercises, with or without numerical results. The student can skip the written part if he/she has passed two written (shorter) exams during the Course. The emphasis is on real "problem solving", that is, the student must be able to model a physical system, solve the relevant equations, and evaluate whether the solution is reasonable (also as far as the order of magnitude of the relevant quantities is concerned). He/she can go to the oral exam if has passed the written exam. In the oral exam, the knowledge of the fundamental laws of electromagnetism is tested, as well as the physical understanding besides the mere use of formulas, the capability of applying this understanding to some different physical cases and, more generally, the critical sense of the student. The oral exam lasts typically 40 minutes.