Mathematics

FRANCESCO PANNARALE GRECO
FRANCESCO PANNARALE GRECO
MASSIMO MASTRODICASA
MASSIMO MASTRODICASA

The aim of the course is to provide the basic theoretical understanding of classical electromagnetism.

Expected results: the ability to lay out and solve standard exercises on electrostatic, magnetostatic, slowly varying electric and magnetic fields.

Acquired knowledge: after passing the exam, the students will be able to profitably follow advanced courses in theoretical physics.

Acquired competences: besides learning the fundamental physics laws of electromagnetism, the students will develop the specific skills needed to address and solve scientific problems via analytical methods, in order to study, model and understand classical electromagnetic phenomena.

General objectives: the aim of the course is to introduce the fundamental theoretical notions for the study and understanding of classical electromagnetism.

Specific objectives
1. Knowledge and understanding: at the end of the course the student will know the fundamental equations of electromagnetism (Maxwell's equations) and will be able to follow advanced courses in theoretical physics.
2. Applying knowledge and understanding: students who have passed the exam will be able to set up and solve standard exercises based on electrostatics, magnetostatics, slowly varying electric and magnetic fields.
3. Making judgements: in addition to learning the fundamental physical laws that govern electromagnetic phenomena, students will develop technical-scientific reasoning skills and analytical problem-solving skills, useful for studying, modeling and understanding the fundamental principles of classical electromagnetism .
These qualities and abilities will be verified periodically during the course with classroom discussions of a wide range of application examples and the resolution of typical problems.
4. Communication: students who have passed the exam will have developed the ability to communicate concepts, ideas and methodologies of classical electromagnetism.
5. Learning skills: the knowledge acquired will allow students to approach, individually or via other courses, more specialized and advanced aspects of classical electromagnetism.

Knowledge of differential and integral calculus, analysis, and linear algebra are essential.

It is essential to have properly absorbed the fundamental laws of dynamics and the concepts of work and potential energy from the course of General Physics I.

1. Electrostatics in vacuum.
* Coulomb's law, the unit of charge in the SI, conservation of charge.
* Definition of electrostatic field E and electrostatic potential V; principle of superposition for E and V.
* Electric dipole, potential and field created by a dipole, mechanical actions on a dipole.

* Electrostatic energy of a system of charges.

* Lines of force and equipotential surfaces.
* Electric field flow; Gauss's theorem; examples of electrostatic field calculations using Gauss's theorem: layer and double layer, undefined wire.

* Equations of electrostatics in vacuum in differential and integral form; Poisson's and Laplace's equations; energy density and pressure of electrostatic field .

2. Conductors in electrostatics.
* Properties of conductors in electrostatic regime, electrostatic induction, hollow conductors, electrostatic screen, image charge method, Capacitors, capacitors connected in series or parallel; energy of a charged capacitor.

3. Electrostatic properties of dielectrics.
* Polarization and dielectric displacement.

* Dielectric constant.
* Maxwell's equations in dielectric media.

4. Electric current.
* Definition of current intensity and current density; continuity equation.
* Stationary currents; resistance of a conductor and Ohm's law.

5. Magnetostatics
* Oersted's and Ampere's discoveries; definition of magnetostatic field; absence of magnetic monopoles.
* Mechanical actions on a magnetic dipole and potential energy of a magnetic dipole.
* Magnetic force on a current-carrying circuit; Lorentz force; motion of a charge in a magnetic field.

* The magnetic field generated by stationary currents: the undefined straight wire, the circular loop and the undefined solenoid.
* The magnetic dipole moment of a current flowing in a loop.

* Magnetic field circuitation, Ampere's theorem; examples of magnetic field calculations using Ampere's circuitation theorem.

* Differential form and integral of the equations of magnetostatics in vacuum.

6. Time-varying magnetic and electric fields.
* Faraday's discoveries and the Faraday-Neumann-Lenz law.
* Case of "cut flux"; alternating f.e.m. generators.
* Autoinduction and mutual induction; closing and opening extracurrents in an RL circuit.
* Hints on alternating current circuits, resonance of an RLC circuit.

* Magnetic energy of a current-carrying solenoid; energy density of the magnetic field.

* Contradiction between Ampere's circuit theorem and conservation of current, introduction of "displacement current".
Maxwell's equations for the electromagnetic field.

7. Electromagnetic waves
* Solutions of Maxwell's equations in vacuum, electromagnetic waves propagating with the speed of light.


8. Lorentz transformations and hints of restricted relativity.

Suggested textbooks
* Fisica II. Elettromagnetismo. Ottica by C. Mencuccini e V. Silvestrni, Ed. CEA
* Fisica Generale - Elettromagnetismo by S. Focardi, I.G. Massa, A. Uguzzoni, M. Villa, Ed. CEA
* Fisica – Elettromagnetismo e ottica by U. Gasparini, M. Margoni, F. Simonetto, Ed. Piccini

Problem solving books di esercizi
* Fisica Vol II by P. Mazzoldi. M. Nigro, C. Voci, Ed. Edises
* Problemi di Fisica generale – Elettromagnetismo ed ottica by M. Nigro, C. Voci, Ed. Cortina

Additional textbooks for deeper insight
* Electricity and Magnetism: Berkeley Physics Course by E. M. Purcell, Ed. Zanichelli
* Introduction to Electrodynamics by D. J. Griffiths, Ed. Prentice-Hall
* Classical Electrodynamics by J. D. Jackson, Ed. John Wiley & Sons

Suggested textbooks
* Fisica II. Elettromagnetismo. Ottica, di C. Mencuccini e V. Silvestrni, Ed. CEA
* Fisica Generale - Elettromagnetismo di S. Focardi, I.G. Massa, A. Uguzzoni, M. Villa, Zanichelli, seconda edizione
* Fisica – Elettromagnetismo e ottica di U. Gasparini, M. Margoni, F. Simonetto, Ed. Piccini

There will be 4 hours a week of frontal lessons and 3 of exercises. During the latter, students will be given time to discuss possible solution strategies amongst themselves, before the problems are solved on the board.

Optional.

The exam includes a written test and an oral test. The written test aims at assessing the student's ability in applying the knowledge acquired during the course. The oral test is aimed at assessing the knowledge and understanding of the topics of the program carried out in class. A mark of at least 16/30 in the written test is a requirement to be admitted to the oral test.

During the semester of the course, two mid-term tests will be held. These will be valid for the purpose of passing the written test. In the event the student passes both tests, it is still possible to attend one of the final written tests and decide whether or not to submit the test, in order to improve one's grade.

Once the written exam has been passed, the grade remains valid only for the corresponding exam session. The student must therefore sustain the oral exam in the same exam session.

1 Electrostatics
A long straight wire, uniformly charged with linear charge density λ = 10^{−6} C/m, constitutes the axis of a long cylindrical shell with internal radius a = 4 cm and external radius b = 8 cm. The shell consists of a homogeneous and isotropic dielectric material with relative dielectric constant εr = 3.
1. Give the expression of the electric field E (modulus, direction and orientation) throughout space.
2. Calculate the value of the work done by an external force to carry a charge q = 10^{−7} C from a point C at a distance c = 10 cm from the wire to a point A at a distance a from it, specifying its sign (positive or negative).
3. Calculate the value of the surface and volumetric density of the polarization charges.
4. How does the answer to question 2 change if the dielectric is not homogeneous but εr = αr, with r distance from the wire axis and α = 4 cm^{−1}?
5. How does the answer to question 2 change if instead of a dielectric shell there is a conductive shell?


2 Magnetism
A square coil has side l = 90 cm, mass m = 80 g and electrical resistance equal to R = 8Ω. Its sides are parallel to the x and y axes and it moves with speed v0 = 1cm/s parallel to the x axis, when, at time t = 0, the magnetic field B⃗ = (B0 + βx)\hat{z} with β = 2.5 T/m is activated. Neglecting the self-induction of the loop, determine:
1. the direction of the induced current and its expression as a function of the speed of the loop at time t;
2. the direction and orientation of the force acting on the loop and its expression as a function of the speed of the loop at time t;
3. the trend of the loop speed as a function of time and its value for t → ∞;
4. the value of the energy dissipated due to the Joule effect by the loop, comparing it with the initial kinetic energy of the coil.