# ELECTRIC AND MAGNETIC FIELDS

OVERVIEW

The undergraduate course “Electromagnetic Fields” introduces and develops basic ideas related to the electromagnetic fundamental laws, to the interaction of electromagnetic fields with matter, to electromagnetic propagation. Its aim is to provide the students with the essential tools for understanding the electromagnetic phenomena and the many practical applications of electromagnetic fields.

## AIMS AND CONTENT

AIMS AND LEARNING OUTCOMES

The course provides the students the basic notions related to electromagnetic fields. During the lectures the electromagnetic fundamental laws, the interaction of electromagnetic fields with matter, the extensions of the laws of conservation of energy and momenta to electromagnetics, and the simplest electromagnetic waves are presented. The course aim is to provide the essential tools for understanding the electromagnetic phenomena and the many practical applications of

electromagnetic fields.

Al the end of the course, the student will be able to describe the main concepts of electrodynamics in the presence of charges in vacuum and in the presence of ponderable media. They will also be able to solve simple electromagnetic problems related to important practical applications.

Teaching methods

All teaching activities are presented by the teacher.

SYLLABUS/CONTENT

1. Course organization, motivation and applications (1.5; 1.5)

2. Some comments on Newtonian, relativistic and quantum physics; the role of classical relativistic electrodynamics in modern physics; some links between classical relativistic and quantum electrodynamics in simple cases; some properties of photons; number and properties of photons involved in many engineering applications (3; 4.5)

3. Recalling some prerequisites: Lorentz force; different models for electric charge distributions; electric current and electric current density; conservation of charge; Maxwell's equations in the presence of charges in vacuum in integral form (2.5; 7)

4. Exercises related to scalar and vector fields, circulations, fluxes, differential operators, international system of units (for electromagnetic quantities); fundamental equations in the presence of charges in vacuum in differential form (3; 10)

5. Electromagnetic fields in the presence of ponderable media:

5.1 Some considerations on the constituents of matter (0.5; 10.5)

5.2 Drawbacks of the approach based on the microscopic Maxwell's equations (0.5; 11)

5.3 The need for macroscopic quantities and relationships among them (0.5; 11.5)

5.4 Conduction current: charge carriers; carrier concentration; its value in solids (conductors, semiconductors, insulators), liquids (electrolytes), gases (e. g., ionosphere, plasma); convection currents; conduction currents in the presence of a single family of carriers; different contributions to the velocity of carriers: thermal, diffusion and drift velocity; Fick's first law; diffusion coefficient; carrier drift; carrier drift in the presence of an electric field; carrier mobility; some important values for the carrier mobility; effects of the temperature on the mobility and on carrier concentration; conductivity; its values in most important materials; unit of measures; first simple constitutive relation for the current density; ideal insulators; superconductors and perfect electric conductors; conduction current in the presence of more families of carriers; other types of current (1.5; 13)

5.5 Electric polarization: exercises related to electrostatics: electric field and scalar potential due to point charges, line, surface or volume distributions of charges, dipoles and ideal dipoles, distributions of dipoles; electric dipole moments; electric dipole moment density (electric polarization or dipole moment per unit volume); surface and volume charge densities equivalent to the distribution of dipoles; some examples of dipole distributions; electronic polarization, atomic-ionic-molecular polarization and polarization due to the orientation of polar molecules; examples: Bohr's atomic model, ethylene molecule, water molecule; macroscopic effect of polarization-charge densities; generalization of Gauss theorem; generalization for time-varying electromagnetic fields; electric displacement; first generalization of Ampere-Maxwell law (5; 18)

5.6 Magnetic polarization: exercises related to magnetostatics: fundamental laws; vector potential; vector potential due to surface or linear current densities; magnetic dipoles; ideal magnetic dipoles; magnetic dipole moment; magnetic dipole moment density per unit volume (magnetic polarization or magnetic dipole moment per unit volume); linear and surface current densities equivalent to the distribution of magnetic dipoles; some examples of magnetic dipole distributions; diamagnetic, paramagnetic and ferromagnetic materials; macroscopic effects of polarization-current densities; final form of Maxwell's equations (integral and differential forms); macroscopic fields; displacement current (3; 21)

5.7 Fundamental equations for the electromagnetic field in the presence of matter (0.5; 21.5)

5.8 Time-harmonic fundamental equations for the electromagnetic field (0.5; 22)

5.9 Exercises: importance of curl and divergence equation (1; 23)

6. Constitutive relationships for ponderable media: linear-non linear; isotropic-anisotropic; dispersive-non dispersive in space and time; homogeneity-inhomogeneity in space and time; examples; integration of time-harmonic Maxwell curl equation and constitutive relations for linear, stationary and spatially non-dispersive media; effective permittivity (3; 26)

7. Boundary conditions at motionless interfaces between different media: conditions for the normal components of the electric displacement, of the magnetic induction and of the Poynting vector; conditions for the tangential parts of the electric and magnetic fields (3; 29)

8. Extension to electromagnetic phenomena of the principle of energy conservation

8.1 Poynting's theorem; physical meaning of the terms appearing in Poynting theorem (3; 32)

8.2 Exercises: exchange between electromagnetic energy and mechanical or thermal energy; Nichols' disk; Joule effect in a cylindrical conductor (2; 34)

8.3 Poynting theorem for time-harmonic fields (2; 36)

8.4 Exercises: power losses due to Joule effect and to dielectric losses; thermal effects in microwave ovens; field amplitudes radiated by isotropic antennas in a lossless and homogeneous medium (3; 39)

9. Conservation of momentum in the presence of charged particles and electromagnetic fields; a comment on the conservation of the angular momentum (2; 41)

10. Uniqueness theorem for the electromagnetic field: general case and time-harmonic case; importance of boundary conditions and of initial conditions; electromagnetic boundary value problems and Cauchy problems (3; 44)

11. Electromagnetic waves

11.1 Electromagnetic fields in simple homogeneous media without charge carriers and impressed current densities: wave equation (1; 45)

11.2 Wave equation in one space dimension: general form of its solution (2; 47)

11.3 Progressive and regressive plane waves; their expressions for a generic direction of propagation (1; 48);

11.4 Electromagnetic plane waves: TEM waves; general expressions for the electric and magnetic fields; speed of light as the velocity of propagation of electromagnetic plane waves; an additional comment on the special theory of relativity (2; 50)

11.5 Other possible waves: spherical waves (1; 51)

11.6 Monochromatic plane waves; wavelength, wavevector, polarization of time-harmonic vectors and vector fields and its practical consequences (polarization division multiplexing, stereoscopic vision, etc.) (3; 54)

11.7 Propagation of plane waves in the presence of absorption: low loss dielectric media and good conductors; attenuation; skin depth; velocity of propagation; some comments on the effects of dispersive media (3; 57)

11.8 Reflection and transmission of a monochromatic plane wave at a plane interface: the case of orthogonal incidence (reflection and transmission coefficients; interference; behavior of the magnitude of the electric field; standing waves; standing wave ratio; measures of the reflection coefficient and of the intrinsic impedance of a material; reflection by an ideal mirror; consequences and applications of interference: Young experiment, simple phased arrays) (3; 60)

RECOMMENDED READING/BIBLIOGRAPHY

- S. Bobbio, E. Gatti, Elementi di elettromagnetismo, Bollati Boringhieri, 1991
- G. Conciauro, L. Perregrini, Fondamenti di onde elettromagnetiche, McGraw-Hill, 2003
- J. D. Jackson, Classical electrodynamics, Wiley, 1999
- D. Pescetti, Elettromagnetismo, Piccin, 1985

The teacher has written the lecture notes for this course. They are available for all students.

## TEACHERS AND EXAM BOARD

**Ricevimento:** Monday, from 5 to 6 p. m., third floor, Via Opera Pia 11a, or by appointment.

Exam Board

MIRCO RAFFETTO (President)

ANDREA RANDAZZO

MATTEO PASTORINO

GIAN LUIGI GRAGNANI

ALESSANDRO FEDELI

## LESSONS

Teaching methods

All teaching activities are presented by the teacher.

## EXAMS

Exam description

The final exam is oral. All students will be asked three questions, of which at least one theoretical and one presented as an exercise.

Assessment methods

At the end of the course the student should show to have understood the basic principle of electrodynamics in the presence of charges in vacuum or in matter and to be able to solve simple problems.