Electric charge. Coulomb’s law. Electric field. Gauss’s law. Electric potential. Conductors and insulators, dieletric materials. Capacitors. Electric current and circuits. Magnetism and Electricity. Magnetic forces: Lorentz force and Laplace's law. Mechanical moments on circuits. Hall effect. Ampere-Laplace and Biot-Savart laws. Magnetic materials. Faraday and Ampère-Maxwell laws. Maxwell's Eq, differential and integral forms. Self-induction. Circuits RL
Mazzoldi, Nigro, Voci - Elementi di fisica, Elettromagnetismo e Onde.
Ed. Edises
Learning Objectives
Acquire basic knowledge on the electrical and magnetic phenomena, through the laws that regulate the static, stationary and time-dependent processes. Introduction to the electromagnetic properties of materials. Ability to deal with, set up and solve electromagnetism problems.
Prerequisites
Mandatory: Physics I
Suggested: Mathematics I
Teaching Methods
Total number of hours for Lectures:32
Total number of hours for Exercises: 24
Further information
Office hours for students:
Torre: Tuesday 11.00-13.00 at LENS, studio 62
Type of Assessment
Written examination based on the solution of a few excercises, and oral examination on the theory of electromagnetism
Course program
Electric force and notes on the first discoveries on electrical phenomena: charge by rubbing and electroscope. Electric charge concept and induction processes. System of units of measurement.
Coulomb's law, superposition principle and definition of electrostatic field (e.s.).
Examples of field calculation: point charges and distributed charges. Lines of force. Millikan experience.
Work and electric potential, electromotive force, potential energy. Calculation of potential and conservative e.s. fields. Potential and potential energy for point and distributed charges. Conservation of energy in electrostatics. Definition of equipotential surface.
Examples of calculation of work and energy, es .. Motion of a charge in a uniform electric field.
Vector operators: gradient, divergence and rotor; gradient, divergence and Stokes theorems. Local properties of the e.s. fileds.
Example of the calculation of the bond potential of the hydrogen atom and calculation of the potential and e.s. of a circular charge distribution, of a loaded disc, of a flat charge distribution and of a double flat charge distribution.
Electric dipole, calculation of the potential. Nabla operator in polar coordinates and calculation of the e.s. dipole. Introduction to the multipole potential.
Forces exerted on an electric dipole. Uniform field: moment of forces and potential energy. Non-uniform field: example of calculation of the resultant of the forces for a simple e.s field.
Example of calculation of the electric fields of two aligned dipoles, energy e.s. of the system and force between the two dipoles. Energy d.e. of two dipoles placed parallel and comparison with the thermal energy to the environment. Straight wire uniformly charged and strength on a dipole placed orthogonally to the wire. Electric dipole oscillating in a field e.s.
Solid angle and vector flow. Eg flow calculation through a spherical cap. Field flow d.e. of point charge. Gauss theorem. Integral and local notation of the theorem, definition of Maxwell's equations for the e.s. Laplacian operator, Poisson and Laplace equations.
Calculation of the field generated by a spherical charge distribution, by a loaded cylinder, by a loaded cylindrical shell and by a spherical charge distribution with density dependent on r, through the Gauss theorem.
Conductors: distribution of surface charges, e.s. field inside and on the surface, Coulomb's theorem. Hollow conductors, complete induction processes and e.s. perfect shield.
Forces between conducting spheres, calculation of the potential.
Capacitors: calculation of the capacity of a spherical and flat capacitor. Circuits with concentrated constants, capacitors in parallel and in series.
Attraction force between the armatures in the case of the insulated capacitor.
Field energy e.s.: calculation for a flat capacitor and general formula.
Example: calculation of the field energy for a spherical and cylindrical capacitor. Various other examples of calculating the e.s. energy.
Magnetism and Electricity, magnetic forces on moving charges: on free charges and on conductors.
Lorentz force: Motion of a charge in uniform B field and its dependence on the direction. Notes on the motion of a charge in a non-uniform magnetic field; magnetic confinement effects. Operating principle of the mass spectrometer.
Magnetic force on a conductor with stationary current: 2nd Laplace's law. Its expression in case of uniform field B, straight wire and flat circuits.
Example: calculation of forces on a flat semicircular loop.
Mechanical moments on circuits: general expression. Calculation for a flat loop in uniform B field. Definition of Magnetic Moment and equivalence principle of Ampere, magnetic dipole. Mechanical moment, potential and work for a magnetic dipole immersed in field B. Hints to the magnetic forces on the magnetic dipole with non-uniform field B.
Example: calculation of the dynamics of a magnetic dipole in field B.
Field B effects in conductors: Hall effect. Definition of the Hall field. Motion of charge carriers and electromotive field.
Other examples on the mass spectrometer. Circuit with current and a mobile side immersed in uniform field B: speed of the side as a function of time. Current in a coil of known magnetic moment immersed in a field not orthogonal to the plane of the coil. Mechanic moment and angle between field and magnetic moment.
Definition of field B sources in stationary conditions. Laplace's first law: its expression and vector meaning, introduction of magnetic permeability. Ampere-Laplace law for a generic circuit. Field B generated by charges in motion. Application of the Ampere-Laplace law for the calculation of B generated by a straight wire, derivation of the Biot-Savart law. Calculation of the B field generated by a circular plane coil, expression on the axis and in space. Connection with the definition of magnetic dipole. Rectilinear solenoid: calculation of the internal field B according to various approximations.
Forces between circuits: attractive and repulsive forces between straight wires crossed by current; definition of current intensity in Ampere.
Demonstration of Ampere's law for the field B circuit, integral and local form. Physical meaning of the dielectric constant and magnetic permeability of the vacuum, speed of light. Gauss's law for field B, flux and divergence of the magnetic field. Maxwell's equations for static-stationary electromagnetism in vacuum.
Magnetic materials; force effects on materials and classification of material types: diamagnetic, paramagnetic and ferromagnetic. Definition of relative and absolute magnetic permeability, of magnetic susceptibility. Introduction to the magnetization vector M, to the vector H and their relationship with the vector B.
Forces on a circular loop traversed by current and placed near a magnet. Spool of mass m constrained to rotate around one of its sides and immersed in uniform field B: current necessary to have a certain rotation and reach an equilibrium position, work of magnetic forces during rotation. Field B in the center of the equilateral triangle formed by 3 wires crossed by the same current, force per unit of length on one of the wires. Applications of Ampere's circuit law.
Microscopic models of magnetization: orbital and spin magnetic moments and their use for diamagnetism and paramagnetism; Weiss domain model for ferromagnetism. Dependence on the field H and the temperature of M in the various materials, Curie's laws; hysteresis cycle in ferromagnetism. Amperian currents and magnetization currents: Ampere's law in magnetic materials. Eq. of Maxwell for static fields in material means and boundary conditions for fields.
Examples: calculation of the magnetic field in a toroidal solenoid. Square spire immersed in the field generated by an undefined wire crossed by current: force acting on the loop, work done to obtain a certain movement of the loop. Calculation of the field generated by a thin metal plate crossed by current and mechanical moment acting on a small magnet placed at a certain distance from the plate. Energy necessary to rotate the atomic dipoles of a paramagnetic gas immersed in uniform B field and comparison with the kinetic energy at room temperature. Solenoid crossed by a current with a thin cylindrical core of ferromagnetic material: fields in the various regions of space, magnetization of the core. Wire crossed by current with external cylindrical metal sheath: calculation of H, B and M in the various regions of the space.
Time-varying fields: experiments on currents induced by field B in conductors. Faraday and Lenz's law; integral and local expression of the law.
Toroidal ring of ferromagnetic material with a winding crossed by current: current necessary to obtain a certain magnetization of the material. Case with 3 winding including two alternating current paths and the other open: maximum ddp across the third winding. Operating principle of the electric motor. Square coil rotating around the median axis immersed in a circular loop with a radius greater than the side of the square coil: application of Felici's law
Induced currents and Lorentz force: calculation of the induced f.e.m on a circuit of variable shape in fixed field B and fixed shape and time-varying field B. Motion of a conductor in the presence of electromagnetic friction. Total charge for induced currents: Felici's laws.
Self-induction phenomenon: definition of the inductance constant L and of the induced f.e.m for inductive effects. Equation for a series RL circuit, calculation of the current for the stationary and transient regime, opening and closing of the circuit.
Square coil rotating around the median axis immersed in a circular loop with a radius much greater than the side of the square coil: calculation of the angular frequency from the knowledge of the average power dissipated. Moment of external forces necessary to maintain rotation. Conductor bar of mass m on inclined metal guides connected to a generator and immersed in uniform field B: f.e.m value to keep the bar in equilibrium, running speed if the generator is short-circuited and power dissipated. Barlow's wheel.
Magnetic field energy: case of RL circuits and definition of energy density, even in the presence of magnetic materials.
Ampere-Maxwell equation: in-applicability of the Ampere law to the case of variable currents, example of the RC circuit. Derivation of the displacement current term from the charge conservation law, reformulation of the field rotor B and its circuitry.
Complete Maxwell equations, in integral and local form.