PHYSICS II

The purpose of the course is to provide basic electromagnetism in vacuum, in the presence of conductors, dielectric and magnetic materials both in stationary conditions and in the presence of time-dependent phenomena, including  electromagnetic waves. At the end of the course the student will be able to solve simple electromagnetism problems starting from Maxwell's equations and constitutive relations related to different materials.

With reference to the topics covered in Physics II, the course will promote the following skills:

- Knowledge and understanding abilities. Inductive and deductive reasoning skills. Ability to schematize a natural phenomenon in terms of scalar and vector physical quantities. Ability to set a simple problem using appropriate relationships between physical quantities (algebraic, integral or differential) and to solve it with analytical methods.

- Ability to apply knowledge and understanding. Ability to apply the knowledge acquired for the description of physical phenomena using rigorously the scientific method. Ability to apply the knowledge acquired to solve simple electromagnetism problems.

 - Making judgments. Critical reasoning skills. Ability to identify the most appropriate methods to analyze critically, interpret and process the data of a problem.

- Communication skills. Ability to present orally, with appropriate language and rigor, a scientific topic, explaining it and illustrating the results.

- Learning skills. Ability to learn the scientific concepts of Physics, necessary to undertake subsequent studies with a high degree of autonomy.

Methods of carrying out the teaching

The teaching is carried out through lectures (for a total of 7 CFU) and exercises (2 CFU) consisting of exercises taken from the reference texts and exams from previous academic years. The exercises will be carried out in cooperative learning mode. The written test can be replaced by two intermediate tests during the course. At the beginning of the course a self-assessment test on the prerequisites required and listed below will be given.

Should the circumstances require online or blended teaching, appropriate modifications to what is hereby stated may be introduced, in order to achieve the main objectives of the course.

Text comprehension. Notions of geometry and vector calculus: distinction between scalar and vector
quantities, sum of vectors, scalar product and vector product. Knowledge of elementary algebra and
trigonometry: solution of first and second algebraic equations degree, trigonometric functions and
goniometric formulas. Differential and integral calculus of one-variable functions. Differential equations of
the first and second order. Newton's laws and equations of motion. Conservative forces and the principle
of conservation of energy mechanics. Translational and rotational dynamics: uniform rectilinear motion
and uniform motion accelerated, angular velocity. Force field.

Attendance to lectures is strongly recommended. Attendance is mandatory to access the ongoing tests
(minimum attendance equal to 65%).

Introduction: Recalls of vector notation and definition of gradient, divergence, and rotor. Divergence theorem and Stokes theorem. [2 hours lecture]

Electrostatic Field: Electrical charges: phenomenology and Coulomb's law. The principle of superposition. An electrostatic field generated by a set of discrete charges. Force lines. Gauss' law. An electrostatic field produced by continuous distributions of charges. Motion of charges in an electrostatic field. Electrostatic potential: Work of the electric force and the electrostatic potential. Electrostatic potential energy, equipotential surfaces. Tension. Electric dipole. Maxwell's equations for the electrostatic field.

[12 hours lectures, 5 hours exercises]

Conductors and electrical conductance: Conductors under equilibrium conditions. Conductance of an insulated conductor. Electrostatic screen. Capacitors in series and parallel connections. Energy stored in a capacitor. 

[4 hours lectures, 3 hours exercises]

Dielectric materials: Phenomenology of dielectrics and polarization vector. Qualitative description of the mechanisms of electronic and orientation polarization. Maxwell's equations in dielectric materials. Continuity properties of the electric fields. Energy of the electric field in the presence of dielectric materials. Microscopic model of the electronic polarizability.

[5 hours lectures, 5 hours exercises]

Direct electric current: Electrical conduction. Electric current. Principle of conservation of charge and continuity equation. Drude model for the conduction and Ohm's law (Joule effect). Resistors in series and in parallel.

[5 hours lectures]

Magnetic field: Magnetic force: phenomenology. Force lines and Gauss's law for the magnetic field. Lorentz law. Force on current-carrying conductors: elementary laws of Laplace. Ampere's principle of equivalence. Magnetic field produced by currents. Ampere's law. Electrodynamic actions between circuits. Maxwell's equations of magnetostatic field. 

[5 hours lectures, 4 hours exercises]

Magnetic media: Phenomenology of magnetic substances and magnetization vector. Maxwell's equations in magnetic media. Continuity properties of the magnetic fields. Energy of the magnetic field in material media. Qualitative discussion of paramagnetism, diamagnetism and ferromagnetism: hysteresis and magnetic shields.

[5 hours lectures, 5 hours exercises]

Time-dependent electric and magnetic fields: Electromagnetic induction, Faraday's and Lenz law. Induced electromotive force. Induction phenomena. Displacement current and Maxwell's Ampere's law. Magnetic energy. Maxwell's equations and electromagnetic waves in a vacuum. Maxwell equations in vacuum in integral and differential forms. 

[5 hours lectures, 4 hours exercises]

Introduction to electromagnetic waves. D'Alembert equation. Symbolic notation. Plane waves. Harmonic waves. Polarization of electromagnetic waves. Energy density of electromagnetic waves, and the Poynting vector intensity. Maxwell's equations in matter and waves in linear media. Qualitative illustration of the phenomena of absorption and dispersion in dielectric materials and conductors

[4 hours lectures, 4 hours exercises]

1) P. Mazzoldi, M. Nigro, C. Voci, Fisica volume II Seconda edizione, EdiSES 2000.
2) Fisica 2, D. Halliday, R. Resnick, K. S. Krane, Zanichelli
3) Edward M. Purcell, La Fisica di Berkley 2, Elettricità e Magnetismo, Zanichelli.

The preparation is assessed through two written tests in progress – a highly recommended method – or with a final written exam, plus a possible oral exam, as described below. Admission to the written test is subject to booking on the Portale Studente platform. For each exam session, a specific range of dates is published in which it is possible to book. At the end of the written test, typically within three days, the test progress is published on STUDIUM, in order to encourage a self-assessment process. The results of the written tests are published on STUDIUM.

- Written tests in progress Two written tests in progress are planned (reserved for students who have attended at least 65% of lessons) to replace the regular written test. The dates (typically mid-November and mid-January) will be communicated during the lessons. Type: resolution, clearly justified and commented, of two problems, whose level of difficulty is similar to the exercises done in class, and a theory topic. Duration: 90 minutes. Evaluation: up to 5 points for each problem well solved and up to 3 points for a correct solution of the theory topic. If the grade obtained in each of the two tests is not seriously insufficient (>8) it contributes to the final evaluation, obtained as the sum of the partial results of the two tests in progress. In the case of a sufficient final grade (>17), the student has the right to have the grade thus obtained recorded (maximum grade obtainable by passing the two ongoing tests 26/30).

Upon request of the student, if the grade obtained (sum of the grades of the two in itinere tests) is >24, the outcome of the in itinere tests can be integrated with an optional oral exam. The outcome of the oral exam contributes to the final evaluation, which can be either better or worse than that resulting from the in itinere written tests alone. The oral exam can be taken in one of the exam sessions provided for in the calendar for the current academic year, upon reservation through the student portal.

The second in-progress test can be taken even if the first one has not been passed. Students who have passed only one of the two in-progress tests (grade >8) will have the opportunity to take the written test in one exam session, tackling only the exercises on the topics of the in-progress test that they did not pass. This will be possible only in a single exam session of the 2024/2025 academic year, after booking the written test on the Portale Studente platform. The score will be the sum of the scores obtained in the in-progress test passed and in the remaining part of the written test. In the case of a sufficient final grade (>17), the student has the right to have the grade thus obtained recorded. The maximum grade that can be obtained with the two tests, in-progress + partial written, is 26/30.

In the event of non-participation in the ongoing tests, of an insufficient result or of refusal of the grade obtained, the student may take the final exam in any of the exam sessions provided for in the calendar upon prior booking.

- Regular written test: Type: resolution, clearly justified and commented, of four problems, whose level of difficulty is similar to the exercises done in class and a theory topic. Duration: 120 minutes. Evaluation: up to 5 points for each problem well solved, up to 6 points for a correct solution of the theory topic. Each regular written test is considered passed if a grade of no less than 18/30 is achieved.

In the case of a sufficient final grade, it is possible to have the grade thus obtained recorded. The maximum grade obtainable by passing the written test is 26/30. Upon request of the student, if the grade obtained is >24, the result of the written test can be integrated with an optional oral test. The result of the oral test contributes to the final grade, which can be either better or worse than that resulting from the written tests alone.

Exam booking through the Smart_Edu platform is mandatory. Non-booked students will not be able to do exams.

The above rules must be understood as useful indications for the student to correct planning and the appropriate preparation for exams, but do not constitute any constraint on the judgment of the examination commission. Learning assessment may also be carried out on line, should the conditions require it, in this case, the duration of the written test may be subject to change.

To ensure equal opportunities and in compliance with current laws, interested students may request a personal interview in order to plan any compensatory and/or dispensatory measures based on educational objectives and specific needs. Students can also contact the CInAP (Centro per l’integrazione Attiva e Partecipata - Servizi per le Disabilità e/o i DSA) referring teacher within their department.

Typical exercise first part of the program:

An electric charge is distributed between two concentric spherical surfaces with rays a = 1cm and b = 30cm uniform density r = 31.2 x 10 -8 in IS units.

1) Derive the electrostatic field in all points of space and the potential by setting the zero of potential to infinity. Calculate the value of the potential at the point r = 35 cm.
2) Calculate how fast an electron initially at rest on the sphere reaches the center of the sphere surface with radius r = 2cm.

Typical exercise second part of the program:

In a square coil with side l = 33 cm, placed on the x-y plane of a Cartesian system, a current flows i = 26 mA. At time t = 0 the half of the coil is subject to the action of a magnetic field B = B z with B = 35 mT

1) Calculate the (vectorial) force that must be applied to the loop for it to be at rest in the system
reference indicated.
2) Calculate the current induced in the loop when it is in motion at t = 0 at constant speed v = 0.5 x m / s and has resistance R = 5 x 10 3 IS units.

Often asked questions:

Maxwell's equations in integral form;
Maxwell's equations in local form;
Electrostatic field and electric field;
Electrostatic induction;
Drude model;
Ohm's law;
Polarization of dielectric materials;
Magnetic materials magnetization;
Constitutive relationships;
Magnetic field;
Wave equation and electromagnetic wave properties;
Polarization of electromagnetic waves.