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In free space Maxwell’s equations become
. E = 0
In free space Maxwell’s equations become
Gauss’s Law
∆
. E = 0 . B = 0
In free space Maxwell’s equations become
No magnetic monopolesGauss’s Law
∆ ∆
. E = 0 . B = 0
x E = - ∂B/∂t
∆
In free space Maxwell’s equations become
No magnetic monopolesGauss’s Law
∆ ∆
Faraday’s Law of Induction
. E = 0 . B = 0
x E = - ∂B/∂t
∆
x B = μo εo (∂E/∂t)
∆
In free space Maxwell’s equations become
No magnetic monopolesGauss’s Law
∆ ∆
Ampère’s LawFaraday’s Law of Induction
∆
= ∂/∂x + ∂/∂y + ∂/∂z
∆
= ∂/∂x + ∂/∂y + ∂/∂z
∆=2 = ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2
∆. ∆
SymbolsE = Electric fieldρ = charge densityi = Electric currentB = Magnetic fieldεo = permittivityJ = current densityD = Electric displacementμo = permeabilityc = speed of lightH = Magnetic field strengthM = MagnetizationP = Polarization
x E = – (∂B/∂t)
∆
x B = μoεo (∂E/∂t)
∆
x E = – (∂B/∂t)
∆
x B = μoεo (∂2E/∂t2) ∆
(∂/∂t)
x B = μoεo (∂E/∂t)
∆
x E = – (∂B/∂t)
∆
x B = μoεo (∂2E/∂t2) ∆
(∂/∂t)
x (∂B/∂t) = μoεo (∂2E/∂t2)
∆x B = μoεo (∂E/∂t)
∆
x E = – (∂B/∂t)
∆
x B = μoεo (∂2E/∂t2) ∆
(∂/∂t)
x (∂B/∂t) = μoεo (∂2E/∂t2)
∆x B = μoεo (∂E/∂t)
∆
x E = – (∂B/∂t)
∆
x B = μoεo (∂2E/∂t2) ∆
(∂/∂t)
x (∂B/∂t) = μoεo (∂2E/∂t2)
∆
x
∆
= – μoεo (∂2E/∂t2)
∆
( x E)
x B = μoεo (∂E/∂t)
∆
x (
∆
x A) =
∆
2A +
∆- ∆
( . A)
∆
x (
∆
x A) =
∆
2A +
∆- ∆
( . A)
∆
2E +∆- ∆
( . E)∆
x (
∆
x A) =
∆
2A +
∆- ∆
( . A)
∆
2E∆-
x (
∆
x A) =
∆
2A +
∆- ∆
( . A)
∆
2E ∆-
= - μo εo (∂E2/∂t2)
x (
∆
x A) =
∆
2A +
∆- ∆
( . A)
∆x (
∆
x E) =
∆
2E +
∆- ∆
( . E)
∆2A ≡ (
∆- ∆. . )A ≡
∆(∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2
= - μo εo (∂E2/∂t2)
x (
∆
x A) =
∆
2A +
∆- ∆
( . A)
∆x (
∆
x E) =
∆
2E +
∆- ∆
( . E)
∆2A ≡ (
∆- ∆. . )A ≡
∆(∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2
= - μo εo (∂E2/∂t2)
x (
∆
x E) =
∆
2E =
∆-
= - μo εo (∂E2/∂t2)
2E =
∆
= μo εo (∂E2/∂t2)
x (
∆
x E) =
∆
x (- ∂B/∂t) =
∆
-(∂/∂t)( x E)
∆
x (∆
x E) =
∆
-(∂/∂t) = μo εo
-(∂/∂t)( x B) ∆
(∂E/∂t) = - μo εo (∂E2/∂t2)
x E = – (∂B/∂t)
∆
x B = μoεo (∂2E/∂t2) ∆
(∂/∂t)
x ∂B/∂t = μoεo (∂2E/∂t2)
∆
x
∆
= – μoεo (∂2E/∂t2)
∆
x E
x B = μoεo (∂E/∂t)
∆
x E = – (∂B/∂t)
∆
x B = μoεo (∂2E/∂t2) ∆
(∂/∂t)
x ∂B/∂t = μoεo (∂2E/∂t2)
∆
x
∆
= – μoεo (∂2E/∂t2)
∆
x E
x B = μoεo (∂E/∂t)
∆
2 E =
∆
- μo εo ∂E2/∂t2
2 E =
∆
(1/c2) ∂E2/∂t2
2 Ψ =∆
- (ώ2/c2) Ψ
ώ = 2πωΨ = ψ e -iώtE → Ψ
2 ψ=
∆
- (ώ/c)2 ψ
ώ/c = 2π/λ
c = ωλ
2 ψ=
∆
- (2π/λ)2 ψ
p = h/λ
ώ/c = 2π/λ2 ψ=
∆
- (2π/λ)2 ψ
p = h / λ
p / h= 1/λ2 ψ=
∆
- (2πp/h)2 ψ
2 ψ =
∆
- (p2/ħ2) ψ
2 ψ =
∆
- (p2/ħ2) ψ
E = T + V
E = p2/2m + V
2m ( E – V ) = p2
2 ψ =
∆
- (2m /ħ2)( E – V ) ψ
2 ψ
∆
+ (2m /ħ2)( E – V ) ψ = 0
. E = 0∆
. B = 0∆
x E = - ∂B/∂t
∆ x B = μo εo (∂E/∂t)
∆In free space Maxwell’s equations become
Ampere’s LawFaraday’s Law of Induction
SymbolsE = Electric fieldρ = charge densityi = electric currentB = Magnetic fieldεo = permittivityJ = current densityD = Electric displacementμo = permeabilityc = speed of lightH = Magnetic field strengthM = MagnetizationP = Polarization
. E = 0
∆
. B = 0
∆
∆
x E = - (∂B/∂t)
∆
xB = μoεo (∂E/∂t)
Maxwell’s Equations
. E = 0
∆
. B = 0
∆
x E = - ∂B/∂t∆
x B = (∂E/∂t) ∆
x (∆
x E) =
∆x (- ∂B/∂t) =
∆-(∂/∂t)( x E)
∆
x (
∆
x E) =
∆
-(∂/∂t) = μo εo
-(∂/∂t)( x B)
∆
(∂E/∂t) = = - μo εo (∂E2/∂t2)
x (∆
x A) = ∆
2A +∆- ∆
( . A)∆
x (
∆
x E) =
∆
2E +
∆- ∆
( . E)
∆
2A ≡ (
∆- ∆
. . )A ≡
∆
(∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2
= - μo εo (∂E2/∂t2)
x (∆
x E) =
∆2E =
∆-= - μo εo (∂E2/∂t2)
2E =
∆
= 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
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 15 30 45 60 75 900 ……………… 3600 x
0
-y
x
v =√ μoεo
1
v =√ μoεo
1
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.
SymbolsE = Electric fieldρ = charge densityi = electric currentB = Magnetic fieldεo = permittivityJ = current densityD = Electric displacementμo = permeabilityc = speed of lightH = Magnetic field strengthM = MagnetizationP = 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
Index
Maxwell's equations concepts
Go Back
Maxwell's Equations
