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Unformatted text preview: PHYSICS 222
Introduction to Classical Physics II
Prof. Ruslan Prozorov
Iowa State University
Fall 2011 LECTURE 19
The Maxwell Equations James Clerk Maxwell (1831–1879)
Born 13 June 1831
Edinburgh, Scotland
Died 5 November 1879 (aged 48)
Cambridge, England
Citizenship United Kingdom
Nationality Scottish In the millennium poll—a survey of the 100 most prominent
physicists — Maxwell was voted the third greatest physicist
of all time, behind only Newton and Einstein.
Maxwell, J. Clerk, “A Dynamical Theory of the Electromagnetic Field”, Philosophical Transactions of the Royal
Society of London, volume 155, pages 459512, 1865, doi: 10.1098/rstl.1865.0008 PHYS222  Lecture 19  Prof. Ruslan Prozorov  Iowa State University 7 October 2011 2 the equations
• Maxwell Equations, in their modern form of four partial
differential equations, first appeared in fully developed
form in his textbook
“A Treatise on Electricity and Magnetism” in 1873.
• Maxwell expressed electromagnetism in the algebra of
quaternions and made the electromagnetic potential the
centerpiece of his theory.
• In 1881 Oliver Heaviside replaced Maxwell’s electromagnetic
potential field by ”force fields” as the centerpiece of
electromagnetic theory. Heaviside reduced the complexity of
Maxwell’s theory down to four differential equations, known now
collectively as Maxwell's Laws or Maxwell's equations.
PHYS222  Lecture 19  Prof. Ruslan Prozorov  Iowa State University 7 October 2011 3 Oliver Heaviside
(18 May 1850 – 3 February 1925) was a selftaught
English
electrical
engineer,
mathematician, and physicist who adapted
complex numbers to the study of electrical
circuits, invented mathematical techniques to
the solution of differential equations (later
found to be equivalent to Laplace transforms),
reformulated Maxwell's field equations in
terms of electric and magnetic forces and
energy flux, and independently coformulated
vector analysis. Although at odds with the
scientific establishment for most of his life,
Heaviside changed the face of mathematics
and science for years to come. PHYS222  Lecture 19  Prof. Ruslan Prozorov  Iowa State University 7 October 2011 4 A Treatise on Electricity and Magnetism PHYS222  Lecture 19  Prof. Ruslan Prozorov  Iowa State University 7 October 2011 5 problem with the Ampere’s law B dl I 0 enclosed when capacitor is
charging/discharging dQ t dV
Q t CV t , I C
dt
dt
there is NO wire inside?
V
where does the current come from?
Q t CV t 0 A 0 AE 0 E
There is electric field (flux)
d
dQ t dE
magnetic field is created by moving charge
ID 0
dt
dt
and by timevarying electric field!
Maxwell postulated the
“displacement current” dE B dl 0 I 0 0 dt PHYS222  Lecture 19  Prof. Ruslan Prozorov  Iowa State University 7 October 2011 6 displacement current Although current is flowing through the capacitor, no actual
charge is transported through the vacuum between its
plates. Nonetheless, a magnetic field exists between the
plates as though a current were present there as well. The
explanation is that a displacement current ID flows in the
vacuum, and this current produces the magnetic field in the
region between the plates according to Ampère's law.
PHYS222  Lecture 19  Prof. Ruslan Prozorov  Iowa State University 7 October 2011 7 elements of differential math ˆ
ˆ
ˆ
x y z
x
y
z  del operator (nabla symbol) 2
2
2 2 2 2 2
x y z divB B  divergence curlB B  Laplacian  curl PHYS222  Lecture 19  Prof. Ruslan Prozorov  Iowa State University 7 October 2011 8 The Maxwell Equations PHYS222  Lecture 19  Prof. Ruslan Prozorov  Iowa State University 7 October 2011 9 in the absence of sources
there is an impressive symmetry of the equations B
E 0
E t
E
B 0 B 0 0
t
PHYS222  Lecture 19  Prof. Ruslan Prozorov  Iowa State University 7 October 2011 10 E produces B, B produces E
Example: At some point P in space, the measured electric field as a function of
time is:
E
P t This E field MUST be accompanied by a B field.
But if the induced B field also changes with time, it induces an E field! This can be selfsustained!
“Perturbation” (E or B field) propagates in space wave
Very special wave, can propagate in vacuum: no medium!
PHYS222  Lecture 19  Prof. Ruslan Prozorov  Iowa State University 7 October 2011 11 the problem is in material
Constitutive relations
In order to apply 'Maxwell's equations', it is necessary to specify the
relations between displacement field D and E, and the magnetic Hfield and
Bfield. These equations specify the response of bound charge and current
to the applied fields (response of the material) and are called constitutive
relations.
The definition of the auxiliary fields: D r , t 0E r , t P r , t H r, t 1 0 B r, t M r, t PHYS222  Lecture 19  Prof. Ruslan Prozorov  Iowa State University 7 October 2011 12 The Maxwell Equations PHYS222  Lecture 19  Prof. Ruslan Prozorov  Iowa State University 7 October 2011 13 example of the use of Maxwell eqs
V1 a) battery V1 R2 V2 R1 R1 B R2 V2 in both
cases
we
will
have
currents
through
resistors b) solenoid with changing field
PHYS222  Lecture 19  Prof. Ruslan Prozorov  Iowa State University 7 October 2011 14 in vacuum (no materials) D r , t 0E r , t H r, t 1 0 B r, t PHYS222  Lecture 19  Prof. Ruslan Prozorov  Iowa State University 7 October 2011 15 isotropic linear materials PHYS222  Lecture 19  Prof. Ruslan Prozorov  Iowa State University 7 October 2011 16 example of the use of Maxwell eqs
electrons in a superconductor (no scattering!) dv
d j ne 2 eE, j nev E
m
dt
m dt d
ne 2
ne 2 B j E dt
m
m t ne 2 ne 2
B 0 j B0 j t m m B
E t B 0 j
B 0
Maxwell equations 0 j B B B B
B 0 ne 2
m B 0 B B 0, 2 London equation PHYS222  Lecture 19  Prof. Ruslan Prozorov  Iowa State University m
0 ne 2 London
penetration
depth
7 October 2011 17 solution of the London equation d 2B
B B z , 0, 0 2 2 B 0 dz d2
d2
d2 2 2 2 2 dx dy
dz B B 0
2 z
B B0 exp so, as expected lim B z 0
z PHYS222  Lecture 19  Prof. Ruslan Prozorov  Iowa State University 7 October 2011 18 ...
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This note was uploaded on 11/14/2011 for the course PHYS 5863005 taught by Professor Meyer during the Fall '09 term at Iowa State.
 Fall '09
 MEYER

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