Physics 504   Electricity & Magnetism    (Spring 2025) 


Course info   | Prerequisites  | Plan of lectures  | Homeworks and Solutions   | Useful Links   | E-mail

COURSE INFO


Room: ARC-204
Time:   Monday,        12:10-1:30 pm
            Thursday,      12:10-1:30 pm
          
Instructor: Sergei Lukyanov
         office: Serin E364
         office phone: (848) 445-9060 
         e-mail: sergei@physics.rutgers.edu (preferred)

Office hours: Friday 10:00 am -12:00 pm
                      Extra meetings can be held by appointment.
                       



The main reference texts: (1) The first part of the course "Special Relativity with basics of relativistic Field Theory" is based on the lecture notes which follows the spirit of L.D. Landau and E.M.Lifshitz "The Classical Theory of Fields", Volume 2

(2) The second part of the course "Application of Classical Electrodynamics" is mostly based on J.D.Jackson "Classical Electrodynamics", 3rd Edition

Additional texts: A. Zangwill, "Modern Electrodynamics", 1st edition
H.Goldstein, C. Poole and J. Safko, "Classical Mechanics", 3rd edition

Homework: (1) There will one homework per 1-2 weeks.

                      (2) Late homework will not be accepted. .
                      The absolute cutoff time for homework is 7 pm due date.

                      (3) Ideally, solutions should be typed (in LaTeX), but
                      handwritten solutions are acceptable as long as they are
                      clearly written. I'll not accept sloppy solutions.

                   (4) Homeworks will be graded and give 30% contribution to your final
                    grade.

Exams: There will be midterm (March 3 (???)) and final (May 8-14) exams.

Final grade: Score % = 30% Homework + 20% Midterm + 50% Final

Students with Disabilities:

If you have a disability, it is essential that you speak to the course supervisor early in the semester to make the necessary arrangements to support a successful learning experience. Also, you must arrange for the course supervisor to receive a Letter of Accommodation from the Office of Disability Services.
http://www.physics.rutgers.edu/ugrad/disabilities.html

Download the course info in PDF format

 
PREREQUISITES  


(I)  Graduate Classical Mechanics at the level Physics 507 or Rutgers challenge exam program:

  • Basic: Lagrangian mechanics, invariance under point transformations, generalized coordinates and momenta, curved configuration space, phase space, dynamical systems, orbits in phase space, phase space flows, fixed points, stable and unstable, canonical transformations, Poisson brackets, differential forms, Liouville's theorem, the natural symplectic 2-form and generating functions, Hamilton-Jacobi theory, integrable systems, adiabatic invariants.

    (II)  Graduate E&M at the level Physics 503 (A. Zangwill, "Modern Electrodynamics", 1st edition; Chapters 1-10) or Rutgers challenge exam program:

  • Basic: Gauss law, differential and integral form Poisson and Laplace equations, Green's theorem, Dirichlet and Neumann boundary conditions, boundary value problems with cylindrical and spherical symmetry, Laplace equation in cylindrical and spherical coordinates, steady current, Biot and Sawart Law, Ampere's law, vector and scalar potentials, Faraday's law, Maxwell's equations.

    •          Download prerequisites

     
    PLAN OF LECTURES


           This is a tentative schedule of what we will cover in the course. It is a subject to change, often without notice.
           These will occur in response to the speed with which we cover material, individual class interests, and possible changes
           in the topics covered. Use this plan to read ahead from the textbooks,

           [1] L.D. Landau and E.M.Lifshitz "The Classical Theory of Fields", Volume 2

           [2] J.D. Jackson ``Classical Electrodynamics'' 3rd edition

           [3] A. Zangwill, "Modern Electrodynamics", 1st edition

           [4] H.Goldstein, C. Poole and J. Safko, "Classical Mechanics", Third edition, Addison Wesley

     

    		 

    		             SPECIAL RELATIVITY WITH BASICS OF
                         RELATIVISTIC FIELD THEORY

     

    		 
     

                       PRELIMINARIES


    •  Space in Classical Physics:  Cartesian space. Euclidean structure. Curvilinear coordinates. Metric-preserving coordinate transformations. Translations of the origin. Proper and improper orthogonal transformations. Euler's theorem. Isometries of E^3. Active and passive points of view.

      Suggested literature: Lecture notes     
                                        Secs.1.2.1-1.2.4,1.7 in [3]
                                        Secs.4.1-4.6 in [4]   
    • Euclidean tensors:  Euclidean vectors and pseudovectors. Levi-Cevita symbol. Cross product. Matrix of finite rotations. Kronecker product. Tensors. Invariant tensors and pseudotensors. Tensors of rank two. Irreducible representations of SO(3). Angular momentum addition. Parity transformations. Irreducible representations of O(3).

      Suggested literature: Lecture notes     
                                        Secs.1.2.5,1.8 in [3]
                                        Secs.4.7,4.8 in [4]  
    • Spacetime in Classical Physics:  Event. Causal structure in classical spacetime. Inertial frames. Galilean transformations. Galilean principle of relativity. Newton's first law.

      Suggested literature: Lecture notes     
                                        Sec.22.2 in [3]

     
    		 
     

                         KINEMATICS OF SPECIAL RELATIVITY

    •  Spacetime in Special Relativity:   Causal structure in Special Relativity. Light cone. Spacetime interval. Proper time. Pseudo-Euclidean (Minkowski) space M^{1,3}. Einstein principle of relativity.

      Suggested literature: Lecture notes     
                                        Secs.1-3 in [1]
                                        Sec.11.1 in [2]
                                        Sec.22.3 in [3]
                                        Sec.7.1 in [4]

            
    • Lorentz group:  Definition. Parity and time reversal transformations. Proper, improper, orthochronous, non-orthochronous Lorentz transformations. General structure of the Lorentz group. Lorentz boosts. Group of proper, orthochronous Lorentz transformations SO^+(1,3).

      Suggested literature: Lecture notes     
                                        Secs.4, 5 in [1]
                                        Sec.11.2 in [2]
                                        Secs.22.4 in [3]
    • Tensors in the Minkowski space:   4-velocity. Covariant and contravariant vectors. Tensors of rank 2. Metric tensor. Inner product in the Minkowski space. Tensors of higher rank in M^{1,3}. Levi-Cevita symbol in M^{1,3}. Pseudotensors.

      Suggested literature: Lecture notes     
                                        Secs.6, 7 in [1]
                                        Sec.11.3, 11.4, 11.6 in [2]
                                        Secs.22.5.1, 22.5.2 in [3]
    • Matrix representations of the Lorentz group:  Rank 2 antisymmetric tensor. Quadratic invariants. Finite dimensional irreducible representations of SO^+(1,3), O^+(1,3) and O(1,3).

      Suggested literature: Lecture notes     
                                        Secs.4, 5 in [1]
                                        Sec.11.2 in [2]
                                        Secs.22.4 in [3]
    		 
     

                         COVARIANT FORM OF MAXWELL'S EQUATIONS


    •  First pair of Maxwell's eqs.:   Fields. Field-strength tensor. Covariant form(s) of the first pair of Maxwell's eqs.

      Suggested literature: Lecture notes     
                                        Secs.23, 26 in [1]
                                        Sec.11.9, 11.10 in [2]
    • Simple physics behind Maxwell's eqs:  Stokes's theorem. Faraday's law of induction. Monopoles. Gauss-Ostrogradsky theorem. Gauss's law. Ampere's law. Displacement current.

      Suggested literature: Lecture notes     
                                        Secs.23-25 in [1]
                                        Secs.1.3, 1.4, 5.1-5.3,5.15, 6.1,6.11,6.12 in [2]
                                        Secs.1.4, 2.1, 2.2 in [3]
    • Second pair of Maxwell's eqs.:  Covariant form. 4-current. The continuity equation.

      Suggested literature: Lecture notes     
                                        Secs.28-30 in [1]
                                        Sec.1.5 in [3]
    • Differential p-forms:   Helmholtz's decomposition theorem. Definition of differential p-forms. Exterior derivative. Closed and exact forms. Poincare lemma.

      Suggested literature: Lecture notes     
                                        Sec.1.9 in [3]
    • 4-potential:  Definition. Bianchi identity. Maxwell's equation in terms of the 4-potential. Gauge invariance. Gauge fixing condition. Lorenz gauge.

      Suggested literature: Lecture notes     
                                        Secs.18 in [1]
                                        Secs.6.2, 6.3 in [2]
                                        Secs.15.3 in [3]
    		 
     

                         VARIATIONAL PRINCIPLE


    •  Poisson's equation in curvilinear coordinates:   Variational principle for Poisson's equation. Laplacian in curvilinear coordinates. Orthogonal coordinates.

      Suggested literature: Lecture notes     
                                        Secs.1.7-1.12 in [2]
    • Variational principle for Maxwell's equations:  From classical mechanics to field theory - continuum mechanics. The principle of least action in relativistic Field Theory. Lagrangian density. Euler-Lagrange equations. The action functional of the electromagnetic field.

      Suggested literature: Lecture notes     
                                        Secs.27, 30, 32 in [1]
                                        Secs.12.7 in [2]
                                        Secs.13.1, 13.2 in [4]
    • Maxwell's equations in curvilinear coordinates (self study):   Tensor fields in curvilinear coordinates. Differentiation. Exterior derivative. Divergency of a vector field. First pair of Maxwell's equations in curvilinear coordinates. The action functional of the electromagnetic field in curvilinear coordinates. Lorenz gauge fixing condition in curvilinear coordinates.
      Suggested literature: Lecture notes     
                                        Secs.81-83, 90 in [1]
    • Functional action for particles in electromagnetic field:  The principle of least action for a free moving particle. Point-like charge in an external field. Covariant form of the equation of motions. Energy conservation law for a charge in a stationary external field. Energy density and energy flux. Poynting vector. Poynting's theorem.
      Suggested literature: Lecture notes     
                                        Secs.8, 9, 15-17 in [1]
                                        Secs.6.7, 12.1 in [2]
                                        Secs.7.9, 7.10 in [4]
    		 
     

                        CONSERVATION LAWS


    • Symmetries: Continuous and discrete symmetries of Classical Electrodynamics. Noether's theorem.

      Suggested literature: Lecture notes     
                                        Secs.6.10 in [2]
                                        Secs.15.1, 15.2, 24.4.2 in [3]
                                        Sec.13.7 in [4]
    • Energy-momentum tensor:   Canonical energy-momentum tensor in Field Theory. Symmetric energy-momentum tensor. Energy-momentum tensor for the electromagnetic field. Conservations of energy and momentum in a local, relativistic invariant field theory. Stress tensor. Energy-momentum tensor of a system of particles.

      Suggested literature: Lecture notes     
                                        Secs.32, 33, 94 in [1]
                                        Secs.6.7, 12.10 in [2]
                                        Secs.13.3, 13.5, 13.6 in [4]
    • Rotational invariance and angular momentum:   4-tensor of angular momentum. The center-of-energy theorem. Pauli-Lubanski 4-vector.

      Suggested literature: Lecture notes     
                                        Secs.14, 32 in [1]
                                        Secs.15.6,15.7 in [3]

     

    		 

    		             APPLICATIONS OF CLASSICAL ELECTRODYNAMICS

     

    		  

                       MAGNETOSTATICS


    • Magnetic moment: Static magnetic field. Vector potential in the Coulomb gauge. Magnetic fields of a localized current distribution. Relation between magnetic and mechanical moments.

      Suggested literature: Lecture notes     
                                        Secs.43, 44 in [1]
                                        Secs.5.3-5.6 in [2]
                                        Secs.11.1, 11.2 in [3]
  • Macroscopic equations: Magnetization. The magnetic field (intensity). Boundary conditions. Relation between magnetic (field) induction and magnetic field (intensity). Methods of solving boundary value problems in magnetostatic.

    Suggested literature: Lecture notes     
                                      Secs.5.8-5.13 in [2]

  • Simple magnetic matter: Magnetic moment in an external magnetic field (torque, force, potential energy). Larmor's theorem. Diamagnetism. Paramagnetism. Curie's law. Exchange interaction.

    Suggested literature: Lecture notes     
                                      Secs.45 in [1]
                                      Secs.5.7 in [2]
    •  
     

                 QUASI-STATIC FIELDS


    • Energy in a magnetic field:  Energy of magnetic matter. Total free energy of a magnetic. Energy of a system of currents. Self- and mutual inductance. Estimation of self-induction for simple circuits.

      Suggested literature: Lecture notes     
                                        Secs.5.16, 5.17 in [2]
  • Quasi-static EM fields in conductors:  Coulomb gauge.Quasi-static approximation. Skin effect.

    Suggested literature: Lecture notes     
                                      Secs.6.3, 5.18 in [2]
                                      Secs.14.5-14.7, 14.10 in [3]
    •  
     

                 ELECTROMAGNETIC WAVES


    •   Waves in vacuum:   Wave equation. Plane EM waves. Monochromatic waves. Helmholtz equation. Doppler effect. Elliptical, linear and circular polarization.

      Suggested literature: Lecture notes     
                                        Secs.46-48 in [1]
                                        Secs.7.1, 7.2 in [2]
                                        Secs.16.1-16.4.4, 16.6 in [3]

    •   Waves in simple matter:  Waves in nondispersive media. Wave impedance. Index of refraction. Reflection and refraction: Snell's law, Fresnel equations, reflection and transmission coefficients, polarization by reflection, Brewster's angle, total internal reflection.

      Suggested literature: Lecture notes     
                                        Secs.7.1, 7.3, 7.4 in [2]
                                        Secs.17.1-17.3 in [3]

    •   Waves in dispersive matter I:   Constitutive relations in a dispersive medium. Kramers-Kronig relations. Lorentz model for dispersion.

      Suggested literature: Lecture notes     
                                        Secs.6.10, 7.5, 7.10 in [2]
                                        Secs.18.51, 18.54, 18.7 in [3]

    •   Waves in dispersive matter II: Plane waves in dispersive media. Phase velocity and group velocity. Conservation of energy in dispersive media: Poynting vector, effective energy density.

      Suggested literature: Lecture notes     
                                        Secs.7.1, 7.3, 7.4 in [2]
                                        Secs.18.3, 18.4, 18.6 in [3]

     

    		 
     

                  RETARDATION AND RADIATION


    •   Fields from moving charges:   Green's functions for the wave equation. Lienard-Wiechert potentials and fields for a point charge. Point charge in uniform motion. Spectral decomposition of the retarded potentials.

      Suggested literature: Lecture notes     
                                        Secs.62-64 in [1]
                                        Secs.6.4, 6.5, 12.11, 14.1 in [2]
                                        Secs.20.1-20.3, 23.1, 23.2 in [3]

    •   Multipole fields and radiation: Fields of a system of charges at large distances. Dipole radiation. Quadrupole and magnetic dipole radiation.

      Suggested literature: Lecture notes     
                                        Secs.66, 67, 71 in [1]
                                        Secs.9.1-9.3 in [2]
                                        Secs.20.5, 20.7 in [3]



      Download syllabus

       
    Homeworks and Solutions  

    		The assignments and solutions are stored in PDF format. The absolute cutoff time for homework is 7pm due date. I'll not accept sloppy solutions.







    Assigned on
    Assignment
    Due Date
    Solutions
    		  
    1. Jan.23, 2025 pdf   Feb.03, 2025 pdf  
    2. Jan.23, 2025 pdf   Feb.10, 2025 pdf  
    3. Jan.30, 2025 pdf   Feb.17, 2025 pdf  
    4. Jan.30, 2025 pdf   Feb.24, 2025 pdf  
    5. Feb.17, 2025 pdf   Mar.06, 2025 pdf  
    6. Feb.17, 2025 pdf   Mar.13, 2025 pdf  

    Midterm:





    March 3, 2025



    Download program and ground rules


    Midterm solutions

    Spring Recess:

    March 15-23, 2025
    7. Mar.6, 2025 pdf   ********* pdf  
    8. Mar.6, 2025 pdf   Apr.3, 2025 pdf  
    9. Mar.24, 2025 pdf   Apr.10, 2025 pdf  
    10. Mar.24, 2025 pdf   Apr.17, 2025 pdf  
    11. Apr.7, 2025 pdf   Apr.24, 2025 pdf  
    12. Apr.7, 2025 pdf   May 1, 2025 pdf  

    Final:





    		May 8-14, 2025, 10am-1pm; ARC-???



    Download program and ground rules



    Grades:
    		Final grades  



    Useful Links  
    1. "Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables",
    M. Abramowitz and I. Stegun

    Course info  |  Prerequisites  |   Plan of lectures  |   Homeworks and Solutions  |   Useful Links  |  E-mail