GENERAL PHYSICS II 2

Module EXERCISES

The course of General Physics II aims to study the laws of electromagnetism and optics, synthesized in Maxwell's equations. The fundamentals of special relativity are introduced, emphasizing the close connection with electromagnetism. The course also covers the initial elements of the classical (non-quantum) theory of radiation. The approach to describing the phenomena studied 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 model a phenomenon in terms of physical quantities, and will be able to critically approach the studied topics, set up a problem, and solve it with analytical methods, rigorously addressing both the mathematical and physical aspects. 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 used. Moreover, the student will be able to clearly and accurately present any topic related to electromagnetism, optics, or special relativity, focusing on the inductive/deductive process that allows conclusions to be drawn from the initial hypotheses.

 

The course includes front lectures (12 credits) and an exercise module (3 credits) which aims to provide students with methodological tools, enabling them to competently tackle general physics problems, particularly in electromagnetism, optics, and relativity.

 

More specifically, the learning outcomes with respect to the Dublin Descriptors for the exercise module are as follows:

 

1. Knowledge and Understanding:

   - Provide a methodical approach to tackling various problems related to electromagnetism, optics, and relativity covered during the course.

   - Provide concrete examples and connections to problem-solving methods used in other areas of physics.

   - Further develop problem-solving skills in the context of general physics problems.

 

2. Applying Knowledge and Understanding:

   - Ability to reason inductively and deductively.

   - Rigorous application of acquired knowledge to describe physical phenomena through the scientific method.

   - Apply the laws of electromagnetism, optics, and relativity to the practical solution of problems, even complex ones.

 

3. Drawing Conclusions:

   - Ability to set up a problem using appropriate mathematical relationships (algebraic equations, differential equations, integrals) between physical quantities and solve these relationships using analytical or numerical methods.

   - Perform calculations on complex problems to derive new results from the laws of electromagnetism (Maxwell’s equations), optics, and relativity.

   - Interpret the results obtained from solved problems and verify their validity.

 

4. Communicative Skills:

   - Clearly and effectively present the fundamental concepts and solutions to the problems worked on through oral and written reports.

   - Actively participate in problem-solving in class and in discussions of possible related topics.

 

5. Learning Skills:

   - Demonstrate the ability to think independently to correctly solve problems proposed in class.

   - Acquire new knowledge and further explore the topics covered during the course.

   - Be able to connect the general concepts of electromagnetism and relativity to recent developments in theoretical and experimental physics.

Front lectures (12 credits)

Exercises (3 credits)

During the first teaching period, the program will cover electrostatics in vacuum and matter, as well as electric currents. The basic concepts of magnetostatics will also be introduced. In the second teaching period, magnetostatics will be further explored, followed by an introduction to special relativity and time-varying electric and magnetic fields. The course will then address magnetism in matter, the study of electromagnetic waves, and electrodynamics, concluding with some fundamentals of optics. If the course is delivered in a hybrid or remote format, necessary adjustments may be made to the previously stated plans to adhere to the Syllabus.

Differential and integral calculus of real functions of a single variable. Vectors in the space and R³ main vector operations. Fundamental concepts of mechanics such as forces, conservative forces, work, kinetic energy, and potential energy. Newton's laws and the differential equation of motion.

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. Electric field circulation. The curl and Stokes' theorem. Work and energy in electrostatics. The electric potential. The potential of a localized charge distribution. The energy of a system of point charges. The energy of a continuous charge distribution. 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 and spherical coordinates. Solutions to Poisson’s equation. Method of image charges. Induction coefficients and potential coefficients. Capacitance of a conductor. Capacitors. Energy stored in a capacitor. Forces between the plates of a capacitor. Electrostatic pressure. Systems of conductors. Electric dipoles. Potential in the far-field approximation. Forces and torques on dipoles. Multipole expansion 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 polarized body of 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 value problems with linear dielectrics. Energy in systems with dielectrics. Dielectric strength.

3 – Electric Currents 

Electric current and current density. Charge conservation and continuity equation. Steady currents. Electrical conductivity and Ohm’s law. Resistivity. Resistance and resistors. Drude’s model of conductivity. Cross-section for rigid sphere collisions. Drift velocity. Mobility. Conductivity. Conductors, semiconductors, insulators. Energy dissipation in current conduction. Joule effect. Electromotive force and voltaic cells. Circuits and circuit elements. Networks with voltage sources. Kirchhoff’s laws. Current sources. Ideal voltage and current generators. Real voltage and current generators. Internal resistance. Slowly varying currents. Charging and discharging of a capacitor. Introduction to electrical conduction in gases.

4 – Magnetostatics 

Magnetic forces. Oersted’s experiment. Lorentz force. Magnetic field. Properties of magnetic forces. Biot-Savart law. The magnetic field of a steady current. Divergence of B. Non-existence of magnetic monopoles. Curl of B. Sources of the magnetic field. Ampère's law. Applications of Ampère's law. Volume and surface current densities. Magnetic field of a circular current loop. Scalar magnetic potential. Vector potential. Helmholtz theorem. Examples of vector potential calculations. Vector potential of a circular loop at a large distance. Magnetic dipole. Magnetic field of a dipole. Forces and torques on magnetic dipoles.

5 – Time-Varying Electric and Magnetic Fields 

Induced electromotive force. Electromagnetic induction. Faraday's law. Applications of Faraday's law. Motional electromotive force. Lenz’s law. 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 substances to magnetic fields. Diamagnetic, paramagnetic, and 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 densities. 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 and magnetic fields. Solutions to the wave equation. Monochromatic plane waves. Polarization. Energy and momentum of the electromagnetic field. Poynting’s theorem. Electromagnetic field momentum. Maxwell’s stress tensor. Energy and momentum of electromagnetic waves. Radiation pressure. Propagation of electromagnetic waves in linear media. Reflection and transmission in normal and oblique incidence. Fundamental laws of geometric optics. Electrodynamics formulation through potentials. Gauge transformations and gauge invariance. Quasi-static approximation. Radiation from point charges.

8 – Fundamentals of Optics 

Nature of light – laws of geometric optics – Fermat’s principle – image formation – mirrors – diopters – thin lenses – light dispersion: prisms – interference of light waves – phasor method – Fraunhofer diffraction – Fresnel diffraction – polarization of light.

 

9 – Electromagnetism and Special Relativity 

Postulates of special relativity. Relativity of simultaneity. Lorentz contraction of lengths and time dilation. Lorentz transformations. Four-vectors. Lorentz transformations in four-dimensional notation. Four-momentum and four-energy. Relativistic invariance of electric charge. Electric field in different inertial frames. Electric field of a point charge moving at constant velocity. Electric field of a point charge starting or stopping. Relativistic interpretation of magnetic force. Magnetic field measured in different inertial frames. Lorentz transformations for electric and magnetic fields in four-dimensional notation.

Main texts: 

1) C. Mencuccini, V. 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 texts suggested for consultation:

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

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

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

 

Texts suggested for exercises: 

7) F. Porto, G. Lanzalone, I. Lombardo, *Problemi di Fisica Generale – Elettromagnetismo 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 involves solving problems within a maximum time of 2 hours. The evaluation of the written test will consider the correctness of the problem-solving approach, the accuracy of numerical calculations and significant figures, and the reasoning supporting the chosen procedure. The minimum score to qualify for the oral exam is 15/30. The oral test will evaluate the depth of content presented and the clarity of language and explanation. It is possible to replace the written exam, and potentially the oral exam, with two midterm tests, one covering electrostatics in vacuum and matter, and electric currents, and the second covering the remaining part of the syllabus. For midterm tests, the same criteria apply as for the regular exams. The minimum passing grade for the written test is 15/30. The first midterm takes place at the end of the first teaching period, during the February exam session. If the written test is passed, students can also participate in the oral test during the same session. Students who pass the first midterm (written or both written and oral) may take the second midterm. The written and oral tests for the second midterm will be offered until the September exam period. Students who pass midterm written exams may be exempted from the regular written test. Students who pass both midterms written, and oral tests will be credited for the entire course, without the need to take the regular exam. The exam may also be conducted online, should conditions make it necessary.

The exam covers all the topics in the course.