GENERAL PHYSICS II 1

Module EXERCISES

The course of General Physics II aims to study the laws of electromagnetism and optics summarized in Maxwell's equations. The foundations of the theory of special relativity are introduced and the close connection with electromagnetism is emphasized. The first elements of the classical theory (in the sense of non-quantum) of radiation are also covered.

The approach to the description of the phenomena covered in the course will be experimental and/or phenomenological and the physical theories will be presented in terms of logical structure, mathematics and experimental evidence.

At the end of the course, the student will have acquired inductive and deductive reasoning skills, will be able to schematize a phenomenon in terms of physical quantities, will be able to critically address the topics studied, to set a problem and solve it with analytical methods, taking care of both the mathematical and physical aspects with due rigor. The student will apply the scientific method to the study of natural phenomena and will be able to critically evaluate analogies and differences between physical systems and the methodologies to be used. Furthermore, he will be able to explain with proper language any topic of electromagnetism, optics or special relativity, focusing on the inductive/deductive process that allows one to reach conclusions from the starting hypotheses.

Classroom lectures (12 credits)

Classroom exercises (3 credits)

During the first teaching period, the part of the program relating to electrostatics, in vacuum and in matter, and to electric currents will be addressed. The first elements of magnetostatics will also be given. In the second teaching period, magnetostatics will be studied in depth and then moved on to the fundamentals of special relativity and time-varying electric and magnetic fields. Then, magnetism in matter, the study of electromagnetic waves and electrodynamics will be addressed. It will conclude with some fundamentals of optics.

If the teaching is taught in mixed mode or remotely, the necessary changes may be introduced with respect to what was previously declared, in order to

respect the planned program and reported in the syllabus.

1 – Electrostatics in vacuum and conductors

Coulomb's law. The electric field. Continuous charge distributions. Field lines, flux, Gauss's law. Divergence of the electric field. Divergence theorem. Applications of Gauss's law. The circulation of the electric field. The rotor and Stokes's theorem. Work and energy in electrostatics. The electric potential. The potential of a localized distribution of charges. The energy of a system of point charges. The energy of a continuous distribution of charges. Energy of the electric field. Conductors: fundamental properties. Conductors in electrostatic fields. Induced charges. Surface charge density. Poisson's equation and Laplace's equation. Solutions of Laplace's equation. Harmonic functions. Boundary conditions in electrostatics and uniqueness theorems. Method of separation of variables in Cartesian coordinates and in spherical coordinates. Solutions of Poisson's equation. Method of image charges. Induction coefficients and potential coefficients. Conductor capacitance. Capacitors. Energy stored in a capacitor. Forces between the plates of a capacitor. Electrostatic pressure. Systems of conductors. Electric dipoles. Potential in the long-distance approximation. Forces and moments of the forces acting on the dipoles. Series expansion of multipoles of the potential.

2 – Electrostatics in dielectrics

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 block of 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. Dielectric strength.

3 – Electric currents

Electric current and current density. Conservation of charge and continuity equation.

Stationary currents. Electrical conductivity and Ohm's law. Resistivity. Resistance and resistors.

Drude model of conductivity. Cross section for collisions of rigid spheres.

Drift velocity. Mobility. Conductivity. Conductors, semiconductors, insulators. Energy dissipation in the conduction of current. Joule effect. Electromotive force and voltaic cell. Circuits and circuit elements. Networks with voltage generators. Kirchhoff's laws. Current sources. Ideal voltage and current generators. Real current and voltage generators. Internal resistance. Currents slowly varying in time. Charging and discharging of the capacitor. Notes on the phenomena of electrical conduction in gases.

4 – Magnetostatics

Magnetic forces. Oersted's experiment. The Lorentz force. Magnetic field. Properties of magnetic forces. The Biot-Savart law. The magnetic field of a stationary current. B. Divergence. Non-existence of magnetic monopoles. B. Rotor. Sources of the magnetic field. Ampère's law. Applications of Ampère's law.

Volume and surface current density. Magnetic field of a circular current loop. Scalar magnetic potential. Vector potential. Helmoltz theorem. Examples of vector potential calculation. Vector potential of a circular loop at a great distance. Magnetic dipole. Magnetic field of a dipole. Forces and moments of forces on magnetic dipoles.

5 – Electric and magnetic fields varying in time

Induced electromotive force. Electromagnetic induction. Faraday's law. Applications of Faraday's law. Electromotive force from induction by motion. 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. LC oscillator. Electrodynamics: displacement current and Maxwell's equations in vacuum. Low-frequency electric oscillations. Alternating currents.

6 – Magnetism in matter

Response of different types of substances 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. Volume and surface magnetization current density. 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 materials. Solution of magnetostatic problems with magnetized materials.

7 – Electrodynamics and electromagnetic waves

Electromagnetic waves. Wave equation for the electric field and the magnetic field. Solutions of the wave equation. Monochromatic plane waves. Polarization. Energy and momentum of the electromagnetic field. Poynting's theorem. Momentum of the electromagnetic field. Maxwell's stress tensor. Energy and momentum of the electromagnetic wave. Radiation pressure. Propagation of electromagnetic waves in linear media. Reflection and transmission in the cases of normal and oblique incidence. Fundamental laws of geometric optics. Formulation of electrodynamics by potentials. Gauge transformations and gauge invariance. Quasi-static approximation. Radiation from point charges.

8 - Fundamentals of Optics

Topics: Nature of light - laws of geometric optics - Fermat's principle - construction of images - mirrors - diopters - thin lenses - dispersion of light: prisms - interference of light waves - phasor method - Fraunhofer diffraction - Fresnel diffraction - polarization of light.

9 - Electromagnetism and the theory of special relativity

Postulates of the theory of special relativity. Relativity of simultaneity. Lorentz contraction of lengths and time dilation. Lorentz transformations. Four-vectors. Lorentz transformations in four-dimensional notation. Energy four-vectors - momentum. Relativistic invariance of electric charge. Electric field in different inertial reference systems. Electric field of a point charge in motion with constant velocity.

Electric field of a point charge that stops or starts moving. Relativistic interpretation of the magnetic force. Magnetic field measured in different inertial reference frames. Lorentz transformations for electric and magnetic fields in four-dimensional notation.

Main books

1) Mencuccini , Silvestrini "Elettromagnetismo e Ottica", Zanichelli

2) P. Mazzoldi - M. Nigro - C. Voci, Fisica, vol. II, EdiSES

3) D.J. Griffiths, Introduction to electrodynamics (IV ed.), Cambridge University Press

Other books:

4) E.M. Purcell, La Fisica di Berkeley: Elettricità e Magnetismo, Zanichelli

5) D. Halliday, R. Resnick, K.S. Krane, Fisica, vol. II (III o IV edizione), Ambrosiana

6) E. Amaldi, R. Bizzarri, G. Pizzella, Fisica Generale, Zanichelli

Testi consigliati per le esercitazioni:

7) F. Porto, G. Lanzalone, I. Lombardo, Problemi di Fisica Generale – Elettrom. e Ottica, EdiSES

8) M. Bruno, M. D’Agostino, R. Santoro, Esercizi di Fisica: Elettromagnetismo, Ambrosiana

The exam consists of a written test and an oral interview. The written test consists of solving problems within a maximum time of 2 hours.

The evaluation of the written test will take into account the correctness of the problem-solving approach, the correctness of the numerical calculations and significant figures, the arguments supporting the procedure followed. The minimum grade for admission to the oral exam is 15/30.

The evaluation of the oral test will take into account the level of depth of the content presented and the propriety of language and presentation.

It is possible to replace the written test, and possibly also the oral exam, with two ongoing tests, the first relating to electrostatics, in vacuum and in matter, and to electric currents and the second relating to the remaining part of the program. Passing the ongoing tests requires passing a written test and possibly an oral exam for each test. The evaluation of the ongoing tests will follow the same criteria described above for the ordinary tests. The minimum grade for passing the written test is 15/30.

The first in itinere test will take place at the end of the first teaching period, in the February exam session. If the written test has been passed, it is also possible to participate in the oral in itinere test, as long as it is in the same session.

Students who have passed the first in itinere test (written exam or written and oral exam) will be able to access the second in itinere test. The written test and the oral test relating to the second in itinere test will be repeated until the September exam session. The student who has passed both written in itinere tests may be exempted from taking the ordinary written test. The student who has passed both written in itinere tests and both oral in itinere tests will instead be recognized for the entire subject without having to take the ordinary exam.

All ordinary written exams have limited validity and it will be necessary to complete the exam, passing the oral exam, within five months of the last written test taken. If the student does not pass the exam within this period, he or she will have to repeat the written test.

.The learning assessment may also be carried out electronically,

should conditions make it necessary.