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In free space Maxwell’s equations become
In free space Maxwell’s equations become
.E
=0
∆
Gauss’s Law
In free space Maxwell’s equations become
=0
∆
Gauss’s Law
∆
.E
.B
=0
No magnetic monopoles
In free space Maxwell’s equations become
=0
∆
Gauss’s Law
x E = - ∂B/∂t
∆
Faraday’s Law of Induction
∆
.E
.B
=0
No magnetic monopoles
In free space Maxwell’s equations become
∆
=0
∆
Gauss’s Law
x E = - ∂B/∂t
∆
Faraday’s Law of Induction
.B
=0
No magnetic monopoles
∆
.E
x B = μo εo (∂E/∂t)
Ampère’s Law
= ∂/∂x + ∂/∂y + ∂/∂z
∆
∆
= ∂/∂x + ∂/∂y + ∂/∂z
2
= ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2
∆= ∆. ∆
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
∆
x E = – (∂B/∂t)
∆
x B = μoεo (∂E/∂t)
∆
x E = – (∂B/∂t)
∆
x B = μoεo (∂E/∂t)
∆
(∂/∂t) x B = μoεo (∂2E/∂t2)
∆
x E = – (∂B/∂t)
∆
x B = μoεo (∂E/∂t)
∆
(∂/∂t) x B = μoεo (∂2E/∂t2)
x (∂B/∂t) = μoεo (∂2E/∂t2)
∆
∆
x E = – (∂B/∂t)
∆
x B = μoεo (∂E/∂t)
∆
(∂/∂t) x B = μoεo (∂2E/∂t2)
x (∂B/∂t) = μoεo (∂2E/∂t2)
∆
∆
x E = – (∂B/∂t)
∆
x B = μoεo (∂E/∂t)
∆
(∂/∂t) x B = μoεo (∂2E/∂t2)
∆
x (∂B/∂t) = μoεo (∂2E/∂t2)
∆
x(
x E) = –
μoεo (∂2E/∂t2)
∆
∆-
x A) =
2A
+
(
∆ ∆
∆
x(
.
A)
∆
2A
∆-
2E
+
+
(
(
∆ ∆
∆-
x A) =
∆ ∆
∆
x(
.
.
A)
E)
∆
∆-
2A
∆-
x A) =
2E
+
(
∆ ∆
∆
x(
.
A)
∆
∆-
2A
∆-
x A) =
2E
+
(
∆ ∆
∆
x(
.
A)
= - μo εo (∂E2/∂t2)
∆
∆ ∆
.
)A ≡
x E) =
∆-
∆
x(
.
2A
+
(
.
A)
(∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2
2E
+
(
∆ ∆
∆
≡(
∆-
∆
2A
x A) =
∆ ∆
x(
.
E)
= - μo εo (∂E2/∂t2)
∆
∆-
∆ ∆
.
)A ≡
x E) =
∆-
∆
x(
.
2A
+
(
.
A)
(∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2
2E
+
(
∆ ∆
∆
≡(
∆-
∆
2A
x A) =
∆ ∆
x(
.
E)
= - μo εo (∂E2/∂t2)
∆
∆-
x E) =
2E
∆-
∆
∆
x(
2E
== - μo εo (∂E2/∂t2)
== μo εo (∂E2/∂t2)
∆
x (- ∂B/∂t) = -(∂/∂t)( x E)
x E) = -(∂/∂t)( x B)
∆
∆
∆
x(
∆
x E) =
∆
x(
∆
∆
-(∂/∂t) = μo εo (∂E/∂t) = - μo εo
(∂E2/∂t2)
∆
x E = – (∂B/∂t)
∆
x B = μoεo (∂E/∂t)
∆
(∂/∂t) x B = μoεo (∂2E/∂t2)
x ∂B/∂t = μoεo (∂2E/∂t2)
∆
∆
x
xE
= – μoεo (∂2E/∂t2)
∆
∆
x E = – (∂B/∂t)
∆
x B = μoεo (∂E/∂t)
∆
(∂/∂t) x B = μoεo (∂2E/∂t2)
x ∂B/∂t = μoεo (∂2E/∂t2)
∆
∆
x
xE
= – μoεo (∂2E/∂t2)
∆
∆
2E
= - μo εo ∂E2/∂t2
∆
2E
2) ∂E2/∂t2
(1/c
=
E →Ψ
Ψ = ψ e -iώt
∆
2Ψ
∆
2 ψ=
- (ώ/c)2 ψ
∆
2 ψ=
- (2π/λ)2 ψ
ώ = 2πω
= - (ώ2/c2) Ψ
c = ωλ
ώ/c = 2π/λ
p = h/λ
2 ψ=
- (2π/λ)2 ψ
ώ/c = 2π/λ
∆
p=h/λ
2 ψ=
2ψ
- (2πp/h)2 ψ
= - (p2/ħ2) ψ
p / h= 1/λ
∆
∆
2ψ
= - (p2/ħ2) ψ
∆
E =T+V
E = p2/2m + V
2m ( E – V ) = p2
∆
2ψ
∆
2ψ
= - (2m /ħ2)( E – V ) ψ
+ (2m /ħ2)( E – V ) ψ = 0
In free space Maxwell’s equations become
∆
x E = - ∂B/∂t
∆
Faraday’s Law of Induction
∆
=0
∆
.E
.B
=0
x B = μo εo (∂E/∂t)
Ampere’s Law
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
Maxwell’s Equations
∆
.E
= 0
∆
.B
= 0
x E = - (∂B/∂t)
x B = μoεo (∂E/∂t)
∆
∆
∆
x E = - ∂B/∂t
∆
=0
∆
.E
.B
=0
x B = (∂E/∂t)
∆
x (- ∂B/∂t) = -(∂/∂t)( x E)
x E) = -(∂/∂t)( x B)
∆
∆
∆
x(
∆
x E) =
∆
x(
∆
∆
-(∂/∂t) = μo εo (∂E/∂t) = = - μo εo (∂E2/∂t2)
∆ ∆
.
)A ≡
x E) =
∆-
∆
x(
.
2A
+
(
.
A)
(∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2
2E
+
(
∆ ∆
∆
≡(
∆-
∆
2A
x A) =
∆ ∆
x(
.
E)
= - μo εo (∂E2/∂t2)
∆
∆-
x E) =
2E
∆-
∆
∆
x(
2E
== - μo εo (∂E2/∂t2)
== μo εo (∂E2/∂t2)
∆
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
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
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.
physics.hmc.edu
image at: www.irregularwebcomic.net/1420.html
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.
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.
. Let’s take a very simple example
y = 4x and y = 3 + x
. Let’s take a very simple example
y = 4x and y = 3 + x
thus
4x = 3 + x
. Let’s take a very simple example
y = 4x and y = 3 + x
thus
4x = 3 + x
3x = 3
. Let’s take a very simple example
y = 4x and y = 3 + x
thus
4x = 3 + x
3x = 3
x =1 and y = 4
. Let’s take a very simple example
y = 4x and y = 3 + x
thus
4x = 3 + x
3x = 3
x =1 and y = 4
Check by back substitution
at: zaksiddons.wordpress.com/.../
Problem 3
Plot on graph paper the function
y = sinx from
x = 0 to x = 360o
y
0
x
-y
0 15 30 45 60 75 900
………………
3600
x
v =
1
√ μoεo
v =
1
√ μoεo
v = 3 x 108 m/s
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.
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
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
Maxwell's
Equations
M
a
x
w
el
l'
s
e
q
u
at
io
n
s
c
o
n
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