# General Physics II

**Academic Year 2022/2023**- Teacher:

**Sebastiano Francesco ALBERGO**

## Expected Learning Outcomes

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

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

At the end of the course, the student will acquire 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; will be able to set up a problem and solve iit, taking care of both the mathematical and physical aspects. Furthermore, he will be able to expose any subject of electromagnetism, optics or special relativity with appropriate language, focusing on the inductive / deductive process that allows to reach conclusions from the starting hypotheses.

## Course Structure

Lessons: 12 credits;

Exercise: 3 credits

`During the first teaching period all the subject related to electrostatics, `

in vacuum and in matter, and electric currents will be addressed. The first elements

of magnetostatics will also be given. During the second didactic period we will deepen

magnetostatics and then move on to the fundamentals of special relativity and time-varying

electric and magnetic fields. Then we will deal with magnetism in matter, the study of

electromagnetic waves and electrodynamics. It will end with the fundaments of optics.

Should the circumstances require online or mixed teaching, appropriate

modifications to what is hereby stated may be introduced, in order to

achieve the main objectives of the course.

## Required Prerequisites

`Differential and integral calculus of one-variable real functions. Vectors in R3 space and `

main operations on vectors. Fundamental concepts of mechanics such as forces, conservative

forces, work, kinetic energy, potential energy. Newton's laws and differential equation of

motion.

## Attendance of Lessons

`Attendance to the course is normally mandatory (consult didactic Regulations of the Course `

of Studies)

## Detailed Course Content

**1 – Electrostatics in vacuum and in conductors**

Coulomb’s law. The electric field. Continuous charge distribution. Field lines., flux, Gauss law. Divergence of electric field. Divergence theorem. Applications of Gauss’s law. The circulation of electric Field. Curl and Stokes’s theorem. Work and energy in electrostatics. Electric potential. The potential of a localized charge distribution. Energy of a point charge distribution. Energy of a continuous charge distribution. Energy of electric field. Conductors: basic properties. Conductors in electric fields. Induced charges. Surface charge density. Poisson equation and Laplace equation. Laplace equation solutions. Harmonic functions. Boundary conditions and uniqueness theorem. Method of variable separation in cartesian and spherical coordinates. Poisson’s equation solutions. Method of image charges. Induction and potential coefficients. Conductor capacity. Capacitors. Energy in capacitors. Forces between capacitor plates. Electrostatic pressure. Conductor systems. Electric Dipole. Large distance potential. Forces and momenta on dipoles. The multipole expansion of electric potential.

**2- Electrostatics in dielectrics**

Dielectrics. Induced dipoles. Alignment of polar molecules. Polarization. Linear dielectrics. Susceptibility. Permittivity. Dielectric constant. Bound charges. Physical interpretation of bound charges. Electric field inside a dielectrics. Gauss’s law in the presence of dielectrics. Electric Displacement **D**. Electrostatic problem in the presence of dielectrics. Boundary conditions. Boundary value problems with linear dielectrics. Energy in dielectric systems. Dielectric rigidity.

**3 - Electric currents**

Electric current and current density. Charge conservation and continuity equation. Stationary currents. Electric conductivity and Ohm law. Resistivity. Resistance and resistors. Drude model of conductivity. Collision cross section for rigid spheres. Drift velocity. Conductivity. conductors, semiconductors, insulators. Energy dissipation of electric currents. Joule effect. Electromotive force and photovoltaic cells. Circuits and circuit elements. Voltage generators. Kirchhoff's laws. Current sources. Ideal voltage and current generators. Real current and voltage generators. Internal resistance. Slowly variable currents over time. Charge and discharge of capacitors. Outline of electrical conduction phenomena 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. The divergence of B. Non-existence of magnetic monopoles. Curl of B. Sources of the magnetic field. Ampère's law. Applications of Ampère's law. Volumic and surface current density. Magnetic field of a circular current loop. Magnetic scalar potential. Vector potential. Helmoltz's theorem. Examples of vector potential calculation. Vector potential of a circular loop at large distance. Magnetic dipole. Magnetic field of a dipole. Forces and moments of forces on magnetic dipoles.

5 - Electric and magnetic fields varying over time

Induced electromotive force. Electromagnetic induction. Faraday's law. Applications of Faraday's law. Motion induced 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. LC oscillator. Electrodynamics: displacement current and Maxwell's equations in vacuum. Low frequency electrical 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. Magnetic intensity **H**. Ampère's law in magnetized materials. Maxwell's equations in matter. Boundary conditions. Qualitative theory of ferromagnetism. Magnets. Linear and non-linear materials. Solving 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 flat waves. Polarization. Energy and momentum of the electromagnetic field. Poynting's theorem. Momentum of the electromagnetic field. Maxwell stress tensor. Energy and momentum of the electromagnetic wave. Radiation pressure. Propagation of electromagnetic waves in linear media. Reflection and transmission in cases of normal and oblique incidence. Fundamental laws of geometric optics. Formulation of electrodynamics through potentials. Gauge transformations and gauge invariance. Delayed potentials. Quasi-static approximation. Point charges in motion: Lienard-Wiechert potentials. Radiation. Radiation of the electric dipole at great distance. Point charge radiation.

**8 - Optics**

Nature of light. Laws of geometric optics. Fermat's principle. Construction of images. Mirrors. Diopters. Thin lenses. Light dispersion: prisms - interference of light - phasor method - Fraunhofer diffraction - Fresnel diffraction - polarization of light

**9 - Electromagnetism and special theory of relativity**

Postulates of the special theory of relativity. Relativity of simultaneity. Lorentz contraction of lengths and dilation of time. Lorentz transformations. Four-vectors. Lorentz transformations in four-dimensional notation. Four-vector energy - momentum. Relativistic invariance of the electric charge. Electric field in different inertial reference systems. Electric field of a point charge in motion with constant speed. Electric field of a point charge that stops or starts. Relativistic interpretation of the magnetic force. Magnetic field measured in inertial reference systems

## Textbook Information

*Main textbooks*:

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

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

*Consultation*:

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

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

*Exercise*:

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

## Course Planning

Subjects | Text References | |

1 | During the first teaching period all the subject related to electrostatics, in vacuum and in matter,and electric currents will be addressed. The first elements of magnetostatics will also be given. | the adopted texts |

2 | During the second didactic period magnetostaticswill be completed and then the fundamentals of special relativity and time-varying electric and magnetic fields will be treated. Magnetism in matter,electromagnetic waves and electrodynamicswill follow. It will end with the fundaments of optics. | the adopted texts |

## Learning Assessment Procedures

The exam consists of a written test and an oral interview. The written test consists in

solving problems within a maximum time of 2 hours. The evaluation of the written test will take into account the correctness of the followed

procedure; the correctness of the numerical calculations, the arguments

supporting the followed procedure. The minimum mark for admission to the oral exam is 15/30. The evaluation of the oral exam will take into account the level of depth of the contents

exposed and the properties of language and exposure. It is possible to replace the written exam, and optionally also the oral one, 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 exam, and optionally an oral exam, for each ongoing test.

For the evaluation of the ongoing tests, the same criteria described above for the

ordinary testswill be followed. The minimum mark to pass each written test is 15/30. The first ongoing test will take place at the end of the first teaching period, in the

February exam session. If the written test is passed, it is also possible to participate

to the oral exam, as long as it is done in the same session. Students who have passed the first ongoing test (written exam, or written and oral exam) will

have access to the second ongoing test. The written test and the oral test relating to the

second ongoing test will be repeated until the September session. The student who has passed

both ongoing written tests may be exempted from taking the ordinary written test. The student

who has passed both written tests and both oral tests will instead have the entire course

recognized, without the need to take the ordinary exam. All ordinary written exams have limited validity. It will be necessary to complete the exam,

passing the oral exam, within five months since written test date. If the student does not

pass the oral exam within this deadline, he will have to repeat the written exam. The verification of learning could also be carried out in telematic way, should conditions

make it necessary.