# 05844 - Electromagnetism (A-L)

• Moduli: Antonio Zoccoli (Modulo 1) Lorenzo Rinaldi (Modulo 2) Roberto Spighi (Modulo 3)
• Campus: Bologna
• Corso: First cycle degree programme (L) in Physics (cod. 9244)

Also valid for First cycle degree programme (L) in Physics (cod. 9244)
• from Sep 25, 2023 to Dec 20, 2023

## Learning outcomes

At the end of the course, the student will have basic knowledge of electromagnetism both in vacuum and in material media, has learned some concepts of vector field analysis and will be able to apply general concepts and fundamental laws of electromagnetism to solving problems.

## Course contents

Microscopic origin of the electrostatic phenomena.

Stable elementary constituents of matter, mass and electric charge. Quantization of electric charge.

General information on the electrostatic field in vacuum

Coulomb's law. Definition of the electric field vector and its aspects: the lines of force, field sources, Gauss's law in differential form. The electric field as a conservative field: the electrostatic potential, the closed path integral and the curl. Density of electrostatic energy associated with the electric field.

Dynamic effects of elementary electrostatic fields

Acceleration of a point-like charge subject to the electric field; 'energy conservation. Electric dipole and the corresponding electric field, electric dipole moment, torque acting on the dipole in an external electric field, the potential energy of the dipole in an external electric field. The dipole approximation.

Electrostatics and conductors

Conductors and insulators. Electric field inside a conductor. Electrostatic induction. Conductors in equilibrium, electric field inside an empty conductor, charge distribution on the surface of the conductor. Uniqueness of the solution of Laplace's equation. Electrostatic capacity. Calculations of capacity for different capacitors: plane, cylindrical and spherical capacitors. Capacitors connected in series and in parallel. Electrostatic screen. The method of images.

The energy of the electrostatic field

Energy of a system of point-like charges and of a continuous distribution of charges. Electrostatic energy stored in a charged capacitor. Localization of energy in the electric field.

Dielectric materials

The electric field in non-conducting materials, the dielectric constant. The polarization of a dielectric material (uniform and non-uniform). Equation of electrostatics in dielectric materials. Linear, isotropic and anisotropic dielectric materials. Discontinuity of the electric field components on the surfaces of separation between two dielectric materials. Electric field inside a cavity. Electrostatic energy in dielectric materials.

Electrical currents

Conduction and electrical current. Definition of current intensity and its measurement units. Carrier current density. Law of conservation of charge: continuity equation. The two Ohm's Law: Resistance and resistivity. The Joule effect. Resistors in series and parallel. Electromotive force from a battery. Kirchoff's laws for electrical networks. RC circuits: charging and discharging of a capacitor through a resistor.

The magnetic field in a vacuum in the stationary case

The magnetic interaction. Lines of force of the magnetic field. Gauss' law for the magnetic field. The II Law of Laplace: magnetic force on a current-carrying conductor. Magnetic force on a moving electric charge. Mechanical moments on planar circuits. Hall effect. The magnetic field on the axis of a coil, the magnetic dipole moment of a coil. Potential energy of the coil in an external magnetic field. Equivalence between a coil traversed by the current and a permanent magnet. Intrinsic and atomic magnetic dipole moments of different materials. Non-separability of the magnetic poles.

The I Law of Laplace (or law of Biot-Savart): magnetic field generated by an electrical current. Calculations of magnetic fields produced by elementary circuits. Ampere's law. Magnetic field in a solenoid. Magnetic fluxes between circuits. Properties of the magnetic field in the vacuum. Vector potential. The transformations of the electric and magnetic fields.

Magnetic fields in the matter

Magnetization of matter. Magnetic permeability and magnetic susceptibility. General equations of magnetostatic. The vector field H. Discontinuity of the fields on the surface of separation between two media magnetized. Fields within a cavity. Diamagnetism, paramagnetism, microscopic interpretation. Ferromagnetism, the magnetization curve interpretation of ferromagnetism.

Time dependent Magnetic and electric fields

Electromagnetic induction and Faraday's law. Lenz's Law and conservation of energy. Physical origin of the induced electromotive force. Applications of Faraday's Law. Inductance, Mutual and Self-induction. Oscillating circuits LC and RL. RLC circuits. Magnetic energy. Mutual induction. Displacement current and Maxwell-Ampere's law.

Maxwell's equations

Discussion of the Maxwell's equations. Electromagnetic waves and energetic aspects of the electromagnetic field. The Poynting theorem.

P. Mazzoldi, M. Nigro, C. Voci, Fisica Vol. 2, Elettromagnetismo - Onde, EdiSES

C. Mencuccini, V. Silvestrini, Fisica II – Elettromagnetismo ed ottica, Liguori editore

S. Focardi, I. Massa, A. Uguzzoni, Fisica Generale - Elettromagnetismo, Casa Editrice Ambrosiana

A. Bertin, N. Semprini Cesari, A. Vitale, A. Zoccoli, Lezioni di Elettromagnetismo, Esculapio Editore (Progetto Leonardo), Bologna.

J. D. Jackson, Elettrodimanica Classica, Zanichelli.

## Teaching methods

The course is organized with lessons in the classroom, in which are presented the basic principles and laws of electromagnetism and optics and wave phenomena, with particular emphasis on the experimental method.

Ample space is also dedicated to the discussion of questions and exercises at the resolution of electrostatic, magnetostatic and electromagnetism. Some practical demonstrations in laboratory are also foreseen.

## Assessment methods

The verification of learning takes place through two intermediate written tests during courses or a final written test with a subsequent oral examination.

Each written test has a duration of 2 hours and during it the use of books, notes or electronic media, is not allowed. The written test consists in solving exercises that aim to ascertain the ability acquired in solving practical problems in the context of the topics addressed during the course and in particular during classroom exercises. To access the oral exam, students must pass the written exam with a minimum grade of 18 out of 30.

The validity of the passed written test is limited to the same exam session, ie the oral exam must be taken in the same session as the written test. The oral exam aims to verify the learning of the fundamental laws of electromagnetism and the acquisition of critical judgment in relation to the solutions of problems.

The final grade takes into account the assessments reported in both tests (written and oral).

## Teaching tools

Laptop. Blackboard.

https://www.bo.infn.it/herab/people/zoccoli/did.html

## Office hours

See the website of Antonio Zoccoli

See the website of Lorenzo Rinaldi

See the website of Roberto Spighi

### SDGs

This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.