Transcript B/∂t - Harry Kroto

James Clerk
Maxwell
Wee Jamie
E
ρ
i
B
εo
J
D
μo
c
H
M
P
=
=
=
=
=
=
=
=
=
=
=
=
Symbols
Electric field
charge density
electric current
Magnetic field
permittivity
current density
Electric displacement
permeability
speed of light
Magnetic field strength
Magnetization
Polarization
Gauss’s Law
q
N
No magnetic monopoles
S
B
∂E/∂t
Ampere’s Circuital Law
i
E
Faraday’s Law of Induction
∂B/∂t
Maxwell took all the semi-quantitative
conclusions of Oersted, Ampere, Gauss and
Faraday and cast them all into a brilliant
overall theoretical framework.
The
framework is summarised in
Maxwell’s Four Equations
Maxwell’s Equations
∆
.E
= 0
∆
.B
= 0
x E = - (∂B/∂t)
x B = μoεo (∂E/∂t)
∆
∆
Feynman on Maxwell'sContributions
"Perhaps the most dramatic moment in the
development of physics during the 19th
century occurred to J. C. Maxwell one day in
the 1860's, when he combined the laws of
electricity and magnetism with the laws of
the behavior of light.
From a long view of the history of mankind —
seen from, say, ten thousand years from now,
there can be little doubt that the most
significant event of the 19th century will be
judged as Maxwell's discovery of the laws of
electrodynamics. The American Civil War will
pale into provincial insignificance in
comparison with this important scientific
event of the same decade.
http://en.wikipedia.org/wiki/File:KL_Kernspeicher_Makro_1.jpg
1nm diametre
10-9 m
1mm diametre
10-2 m
ca 10,000,000 smaller
These equations are a bit complicated
and we are not going to deal with them
in this very general course. However
we can discuss arguably the most
important and at the time most amazing
consequence of these equations.
N
x B = μoεo ∂E/∂t
∆
N
S
x E = - ∂B/∂t
∆
As equations are combined – for instance
when one has two equations in two
unknowns one can juggle the equations and
obtain two new equations each involving only
one of the unknowns and so solve them.
εo = dielectric constant or pemittivity
μo = permeability
μoεo =
1
2
c
Maxwell’s Equations - it’s hard to get a shot
of all four at once - it’s hard to get my arm in
the right position for any camera to get
Ampere’s circuital law.” Weldon
Harry Kroto 2004
Harry Kroto 2004
image at: talklikeaphysicist.com/2008/11/
Harry Kroto 2004
As a result, the properties of light were partly
unravelled -- that old and subtle stuff that is
so important and mysterious that it was felt
necessary to arrange a special creation for it
when writing Genesis. Maxwell could say,
when he was finished with his discovery, 'Let
there be electricity and magnetism, and there
is
light!'
"
Richard Feynman in The Feynman Lectures
on Physics, vol. 1, 28-1.
Maxwell took all the semi-quantitative
conclusions of Oersted, Ampere, Gauss and
Faraday and cast them all into a brilliant
overall theoretical framework.
The
framework is summarised in
Maxwell’s Four Equations
2-D Array of Ni6 clusters
with Prashant Jain Naresh Dalal and Tony Cheetham
Calculus
Differentiation
Calculus
Differentiation
dy/dx = y
Calculus
Differentiation
dy/dx = y
y = ex
Calculus
Differentiation
dy/dx = y
y = ex
eix = cosx + i sinx
dsinx/dx = cosx
dsinx/dx = cosx
and
dcosx/dx = - sinx
dsinx/dx = cosx
and
dcosx/dx = - sinx
thus
d2sinx/dx2 = -sinx
These equations are a bit complicated
and we are not going to deal with them
in this very general course. However
we can discuss arguably the most
important and at the time most amazing
consequence of these equations.
∆
x E = - (∂B/∂t)
.B =
∆
0
∆
.E=
0
x B = μoεo (∂E/∂t)
∆
∆
x (- ∂B/∂t) = -(∂/∂t)( x E)
∆
x (- ∂B/∂t) =
x E) = -(∂/∂t)( x B)
∆
∆
∆
x(
x E) =
∆
x(
∆
∆
x (- ∂B/∂t) = -(∂/∂t)( x E)
x E) = -(∂/∂t)( x B)
∆
∆
∆
x(
∆
x E) =
∆
x(
∆
∆
-(∂/∂t) = μo εo (∂E/∂t) = = - μo εo (∂E2/∂t2)
Lesson 19 - Ampere's Law As Modified by Maxwell
I.
Capacitor Problem
C
In the figure below, we attempt to apply Ampere's law for a wire
leading to a
capacitor using the curve, C. IcIc
Although it is easy to write Ampere's Law, we find that we have a
paradox since
the value for the integral depends on the current that
penetrates any surface
bounded by the curve.
For a circular membrane, the current passing through the
surface is Ic.
For a surface that wraps around the capacitor, we have no
current penetrating the surface.
In 1865, James Clerk Maxwell, one of the great physicists of all
times, solved this paradox and developed electromagnetic theory (one of
the major branches of
physics).
II.
Ampere's Law As Modified By Maxwell
The current flowing through the wire supplies the charge on the
plates of the
capacitor that produces the electric field across the
capacitor plates. From our previous work with parallel plate capacitors, we
have
From the definition of current, we have
Maxwell's contribution was to imagine that the changing electric flux
between the
capacitor was equivalent to a physical current as far as
the creation of a magnetic field. He called this "fictitious" current the
displacement current.
To remove the paradox, Maxwell equated the displacement current
to the current
in the wire and modified the right hand side of Ampere's
Law to include the sum
of the real current and the displacement current.
In our work, there is no difference for our parallel plate capacitor
between the
partial derivative of the electric flux with respect to time
and the full time derivative of the electric flux. However, Maxwell using
more powerful
mathematical techniques solved the problem in general
thereby showing that it is the partial derivative of the electric flux with
respect to time. Thus, we have
written our final result so that it will be
correct for all problems.
IMPORTANT: This incredible result states that there is a second way to
create a circulating magnetic field: A time varying electric flux!! We will
return later in the course to this wonderful result and its importance in
communications.
at: zaksiddons.wordpress.com/.../
N
x B = μoεo ∂E/∂t
∆
N
N
S
physics.hmc.edu
image at: www.irregularwebcomic.net/1420.html
LAW
DIFFERENTIAL FORM
INTEGRAL FORM
Gauss' law for
electricity
Gauss' law for
magnetism
Faraday's law
of induction
Ampere's law
NOTES: E - electric field, ρ - charge density, ε0 ≈ 8.85×10-12 - electric permittivity of free space, π ≈ 3.14159,
k - Boltzmann's constant, q - charge, B - magnetic induction, Φ - magnetic flux, J - current density, i - electric
current,
c ≈ 299 792 458 m/s - the speed of light, µ0 = 4π×10-7 - magnetic permeability of free space, ∇ - del operator (for a
vector function V: ∇. V - divergence of V, ∇×V - the curl of V).
at: www.physics.hmc.edu/courses/Ph51.html
The
lineLaw
integral
of the electric field around a closed loop is equal to the ne
Faraday's
of Induction
Ampere's
law
Application
Gauss'
Faraday's
law,
law
electricity
magnetism
to voltage
generation
in a coil voltage or emf in the loop, so Fa
This line
integral
is equal
to the generated