Essentials to memorize
You need to memorize the following essentials
because they cannot be found on the formula sheet:
Gradient, divergence and curl
Coulomb Law (point charge, (r)
Electric field (point charge, (r)
Gausss Law (in vacuum and matter)
E
Todays lecture
XII. Electrodynamics and Relativity
Transformation of the fields in different inertial
systems
Next time:
Summary of entire class
How do the fields transform?
Main assumption: Charge is invariant (independent of
its motion).
Consider the un
Todays lecture
XII. Electrodynamics and Relativity
Einsteins postulates
Einsteins Gedankenexperiments
The Lorentz Transformations
Next time:
Relativistic electrodynamics:
Magnetism as a relativistic phenomenon
Transformation of the fields in different ine
Todays lecture
VII. Electrodynamics
How Maxwell fixed the last of the Maxwell
equations
Maxwell equations in Matter
Boundary conditions
Summary/overview over whole semester
What Maxwell did
Maxwells contribution was to rewrite J
in terms of
the electric f
Todays lecture
VII. Electrodynamics
Energy in fields
Maxwell equations
Next time:
Maxwell equations in Matter
Summary
What does the current density depend on?
For most materials J is proportional to the force (on each
charge). The proportionality factor i
Todays lecture
VII. Electrodynamics
Electromotive force
Motional emf
examples
Electromagnetic induction
Joule heating law
Electric field E does work on the moving charges,
The work does not increase the kinetic energy of the charges
(constant currentscons
Todays lecture
VII. Electrodynamics
Dynamics
Ohms law
Conductivity, resistivity and resistance
Magnetostatics and electrostatics
Magnetostatics
0 I ( r ' ) ud
B(r ) =
dl '
2
4
d
B = 0
B da = 0
S
B = 0 J (r )
B dl = 0 I enclosed
P
B
B = A
A
m := I a
M
Re-run: Bound and free currents:
0
J b (r ' )
K (r ' )
A(r ) =
d ' + 0 b
da '
d
4
4 S d
volume current J b = M
K b = M un surface current
The magnetic potential in all cases can be expressed as
A(r ) = Abound volume + Abound surface
Magnetic potenti
Todays lecture
VI. Magnetostatics in matter
Bound currents
Auxiliary field H
Boundary conditions for H and B
Magnetic Susceptibility and Permeability
Bound currents:
summing over the dipole moments
Can the magnetic potential be expressed as combination of
Todays lecture
VI. Magnetostatics in matter
Examples
Magnetic dipole moment
Magnetization M
Magnetic fields in matter
Study of the magnetic (and electric fields) fields in matter
is important in order to understand
Interaction electromagnetic radiation -
Todays lecture
V. Magnetostatics (in vacuum)
Gauge transformations
Magnetostatic boundary conditions
Examples
Last session
B = 0
B da = 0
S
magnetic fields of steady currents Ampres law
B = 0 J (r )
B dl = 0 I enclosed
P
Electrostatic:
Magnetostatic:
Todays lecture
V. Magnetostatics (in vacuum)
Divergence of magnetic field B
Curl of magnetic field B
The magnetic vector potential
Divergence of B
For a volume current density we found:
0
J ( r ' ) ud
B(r ) =
d '
2
4 V
d
Separation
vector
d = r r'
What
Todays lecture
V. Magnetostatics (in vacuum)
Magnetic field
Lorentz force
Current and current densities
Continuity equation
The Law of Biot-Savart
Magnetic field B and Lorentz Force
In electrostatics, we studied forces and fields of stationary
charges.
S
Todays lecture
IV. Electrostatics in matter
Energy in dielectric systems
Examples
Energy in dielectric systems
In general one has to apply more energy in order to
obtain a certain potential when a dielectric material is
involved (because of the shielding
Todays lecture
IV. Electrostatics in matter
Electric displacement
Gausss law in Dielectrics
Susceptibility, Permittivity, Dielectric constant
Gausss law in dielectrics
The net effect of a dielectric on the total field E is to
increase D inside by the amou
Todays lecture
II. Electrostatics in vacuum
The problems with continuity of the fields E, V
Work and Energy in Electrostatics
Poissons and Laplaces equations
How do the fundamental equations for look like for V?
E =
2
= V = V
E = 0
= V
0
(
2
V =
)
(
Daily Menu
II. Electrostatics in vacuum
Application for Gausss law
Coaxial problem
Curl of static electric field
Electrostatic potential
The divergence of E
Our 1st Maxwell equation: Gausss Law
Differential form
E =
0
Integral Form
Qenclosed
E da =
S
Todays lecture
II. Electrostatics
Continuous charge distributions
Examples
Field lines
Gausss Law
Divergence of electric field
Continuous Charge distributions
So far point charges were considered.
If the charge is distributed continuously
over a region we
Todays lecture
The Dirac Delta function in one dimension
The Dirac Delta function in three dimension
I. Classification of vector fields
Circulation and flux
Harmonic, solenoidal and conservative fields
Helmholtz theorem
II. Electrostatics
Superposition
Co
Todays lecture
III. Integral calculus:
Line integrals
Surface integrals
Volume integrals
Fundamental theorem of calculus
Fundamental theorem for gradients
Fundamental theorem for divergences
Fundamental theorem for curls
Integration by parts
Integral calc
Todays lecture: Vector algebra
Scalar, vectors (and fields)
Component form of vectors
Equality of vectors
Addition of vectors
Negative Vector and Subtraction of vectors
Geometrical representation: how to shift vectors
Multiplication by a Scalar
Magnitude
Essentials to memorize
You need to memorize the following essentials
because they cannot be found on the formula sheet:
Gradient, divergence and curl
Coulomb Law (point charge, (r)
Electric field (point charge, (r)
Gausss Law (in vacuum and matter)
E
EP 225.3 WAVES, FIELDS AND OPTICS
FINAL EXAMINATION
April 13, 2009
Solutions
1. A thin lens made of glass with ng = 1:5 has a focal length of 10 cm in air. If it is
placed in a liquid, it becomes a divergent lens with f = 50 cm. Find the index of
refracti