27215 - General Physics 2

Learning outcomes

At the end of the course, the student acquires the basic knowledge of electromagnetism and optics of wave phenomena. In particular, will be able to address and solve classical problems of electrostatics, magnetostatics, electromagnetism, waves and optics.

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 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 vector.

Oscillators and Waves
Oscillatory phenomena: free oscillations, damped and forced. The phenomenon of resonance. Elastic and absorption amplitudes. Introduction to wave phenomena. Progressive and stationary, longitudinal, transverse, plane and spherical waves. Phase and Group velocity. D'Alembert equation. Solution of the D'Alembert: progressive and regressive waves. Harmonic waves.
Dispersion relation, relationship between wavelength and frequency. Phase velocity. Fourier analysis. Study of a progressive wave. Energy, power and intensity of a wave. Impedance of a medium. Reflection and transmission of a wave in media of different impedance. Energy balance. Superposition principle. Superposition of a progressive wave and a regressive. Stationary waves. Rope with two ends constrained. Antinodes and nodes. Normal modes and harmonics. The harmonic frequencies. Stationary waves as a series of harmonics. Musical notes. Propagation of sound in air. Speed of sound. Power, intensity and energy carried by sound waves. The Decibel and the human ear. Stationary waves in a column of gas. The principle of superposition and beats. Harmonic waves in 3 dimensions. Plane waves. D'Alembert equation in space. Spherical and cylindrical waves. Complex representation of the waves. Wave packets and Fourier analysis. Phase and group velocity in dispersive media. Doppler Effect sound and light. Shock waves and Mach number.

Electromagnetic waves
Equation of electromagnetic wave derived from Maxwell's equations. Transversality and properties of electromagnetic waves. Speed of light in vacuum and in a medium. Impedance. Representation of an electromagnetic wave. Polarization: linear, elliptical and circular. Energy carried by an electromagnetic wave. Poynting's theorem and Poynting vector. Intensity of electromagnetic waves. Radiation pressure, momentum of an electromagnetic wave. Spectrum of electromagnetic waves and visible light. Propagation in a dielectric: wave equation and  the dispersion relation. Complex refractive index. Descartes Laws of refraction and reflection. Snell's law. Total internal reflection and critical angle. Fermat's principle and Snell's law. Dispersion in a prism. The rainbow: primary and secondary bow, dark band of Alexander. Wave interpretation of the laws of Descartes through the continuity of the electric field. Evanescent wave and total reflection.
Intensity of reflected and refracted waves. The Fresnel formulas. Reflectance and transmittance. Brewster angle. Methods to polarize a wave: polaroid. Reflection and refraction at normal incidence to the surface. Introduction to the phenomenon of light scattering. Dispersion in dielectrics. Imaginary refractive index, absorption and propagation of an electromagnetic wave. Dependence of the refractive index from the frequency. Group velocity of an electromagnetic wave in a dielectric. Propagation in a metal. Wave equation in a metal and corresponding solution.

Interference and diffraction
Principle of Huygens-Fresnel. Introduction interference. Sum of the amplitudes of two waves with the phasor and the symbolic methods. Interference of light and electromagnetic waves: Young's experiment. The intensity distribution on the screen. Optical path. Terms of polarization. Interference with lenses. Interference of thin foils and thin wedge.
Interference by N coherent light sources. Primary and secondary maxima. Diffraction: Fresnel and Fraunhofer diffraction. Light intensity on a screen. Diffraction from a circular hole and from an object. Babinet principle. Resolving power. Diffraction grating. Dispersive and resolving power of a grating. Spectroscopy with diffraction grating. The Michelson-Morley experiment.

Geometric optics
Introduction geometric optics. Light Rays and laws of Descartes. Transparent and reflective  surfaces (mirrors and lenses). Objects and images. Paraxial approximation. Properties of concave and convex mirrors. Focus and focal distance. Magnification factor. Plane mirrors. Concave and convex lenses and their properties: equation of thin lenses and their properties. Convergent and divergent lens. Magnification. Aberrations. Optical instruments. The human eye.

 

 

Readings/Bibliography

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

C. Mencuccini, P. Silvestrini, Fisica II, Elettromagnetismo - 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.

 

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, electromagnetism and optics.Some didattic laboratory demonstration are foreseen.

Assessment methods

The assesment of learning takes place through intermediate written tests during courses or a final written exam and an a subsequent oral examination. During the written tests is not allowed the use of books, notes, calculators, electronic media.

The written test is designed to verify the skills acquired in solving problems concerning the arguments addressed in the course. A positive judgment allows to access to the oral test. The validity of the written test is limited to the same examination session. Both the written and oral tests have the purpose of verifying the learning of the basic laws of electromagnetism and optics of wave phenomena, and the acquisition of critical judgment in relation to the solutions of the problems. The final vote takes into account of votes reported in both parts.

Teaching tools

None

Links to further information

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

Office hours

See the website of Antonio Zoccoli

See the website of Lorenzo Rinaldi