Post on 29-Dec-2015
Maxwell’s EquationsChapter 32, Sections 9, 10, 11
Maxwell’s Equations
Electromagnetic WavesChapter 34, Sections 1,2,3
The Equations of Electromagnetism (at this point …)
€
E∫ • dA =q
ε0
€
B∫ • dA = 0
€
E∫ • dl = −dΦB
dt
€
B∫ • dl = μ0I
Gauss’ Law for Electrostatics
Gauss’ Law for Magnetism
Faraday’s Law of Induction
Ampere’s Law
1
2
The Equations of Electromagnetism
€
E∫ • dA =q
ε0
€
B∫ • dA = 0
..monopole..
?...there’s no magnetic monopole....!!
Gauss’s Laws
4
The Equations of Electromagnetism
€
E∫ • dl = −dΦB
dt
€
B∫ • dl = μ0I
3
.. if you change a magnetic field you induce an electricfield.........
.. if you change a magnetic field you induce an electricfield.........
.......is the reversetrue..?
.......is the reversetrue..?
Faraday’s Law
Ampere’s Law
Ampere’s Law EB
€
B∫ • dl = μ0I
Look at charge flowing into a capacitor
Here I is the current piercing the flat surface spanning the loop.
E
Ampere’s Law
For an infinite wire you can deform the surface and I still pierces it. But something goes wrong here if the loop encloses one plate of the capacitor; in this case the piercing current is zero.
B
Side view: (Surface is now like a bag:)
EB
€
B∫ • dl = μ0I
Look at charge flowing into a capacitor
Here I is the current piercing the flat surface spanning the loop.
E
It must still be the case that B around the little loop satisfies
where I is the current in the wire. But that current does not pierce the surface.
B
EB
€
B∫ • dl = μ0I
Look at charge flowing into a capacitor
What does pierce the surface? Electric flux - and that flux is increasing in time.
EB
EB
Look at charge flowing into a capacitor
€
q = ε0EA
I =dq
dt= ε0
d(EA)
dt
I = ε0
dΦE
dt
Thus the steady current in the wire produces a steadily increasing electric flux. For the sac-like surface we can write Ampere’s law equivalently as
€
B∫ • dl = μ0ε0
dΦE
dt
EB
EB
Look at charge flowing into a capacitor
The best way to write this result is
Then whether the capping surface is the flat (pierced by I) or the sac (pierced by electric flux) you get the same answer for B around the circular loop.
€
B∫ • dl = μ0I + μ0ε0
dΦE
dt
€
B∫ • dl = μ0I + μ0ε0
dΦE
dt
EB
Maxwell-Ampere Law
This result is Maxwell’s modification of Ampere’s law:
€
B∫ • dl = μ0I + μ0ε0
dΦE
dt
€
B∫ • dl = μ0I + μ0ε0
dΦE
dt
Can rewrite this by defining the displacement current (not really a current) as
€
Id = ε0
dΦE
dtThen
€
B∫ • dl = μ0(I + Id )
EB
Maxwell-Ampere Law
This turns out to be more than a careful way to take care of a strange choice of capping surface. It predicts a new result:
A changing electric field induces a magnetic field
This is easy to see: just apply the new version of Ampere’s law to a loop between the capacitor plates with a flat capping surface:
x
x x x x
x x x x x
x x
B
€
B∫ • dl = μ0ε0
dΦE
dt
B 2πr( ) = μ0ε0 πr2
( )dE
dt⇒ B =
μ0ε0r
2
dE
dt
Maxwell’s Equations of Electromagnetism
Gauss’s Law for Electrostatics
Gauss’s Law for Magnetism
Faraday’s Law of Induction
Ampere’s Law
€
E∫ • dA =q
ε0
€
B∫ • dA = 0
€
E∫ • dl = −dΦB
dt
€
B∫ • dl = μ0I + μ0ε0
dΦE
dt
Maxwell’s Equations of Electromagnetism
Gauss’s Law for Electrostatics
Gauss’s Law for Magnetism
Faraday’s Law of Induction
Ampere’s Law
€
E∫ • dA =q
ε0
€
B∫ • dA = 0
€
E∫ • dl = −dΦB
dt
€
B∫ • dl = μ0I + μ0ε0
dΦE
dt
These are as symmetric as can be between electric and magnetic fields – given that there are no magnetic charges.
Maxwell’s Equations in a Vacuum
Consider these equations in a vacuum: no charges or currents
€
E∫ • dA =q
ε0
€
B∫ • dA = 0
€
E∫ • dl = −dΦB
dt
€
B∫ • dl = μ0I + μ0ε0
dΦE
dt
€
E∫ • dA = 0
€
B∫ • dA = 0
€
E∫ • dl = −dΦB
dt
€
B∫ • dl = μ0ε0
dΦE
dt
Consider these equations in a vacuum: no charges or currents
€
E∫ • dA =q
ε0
€
B∫ • dA = 0
€
E∫ • dl = −dΦB
dt
€
B∫ • dl = μ0I + μ0ε0
dΦE
dt
€
E∫ • dA = 0
€
B∫ • dA = 0
€
E∫ • dl = −dΦB
dt
€
B∫ • dl = μ0ε0
dΦE
dt
These integral equations have a remarkable property: a wave solution
Maxwell’s Equations in a Vacuum
Plane Electromagnetic Waves
x
Ey
Bz
E(x, t) = EP sin (kx-t)
B(x, t) = BP sin (kx-t) z
j
c€
B∫ • dl = μ0ε0
dΦE
dt
E∫ • dl = −dΦB
dt
This pair of equations is solved simultaneously by:
as long as
€
E p Bp =ω /k =1/ ε0μ0
Static waveF(x) = FP sin (kx + ) k = 2 k = wavenumber = wavelength
F(x)
x
Moving waveF(x, t) = FP sin (kx - t) = 2 f = angular frequencyf = frequencyv = / k
F(x)
x
v
x
v Moving wave
F(x, t) = FP sin (kx - t )
At time zero this is F(x,0)=Fpsin(kx).
F
x
v Moving wave
F(x, t) = FP sin (kx - t )
At time zero this is F(x,0)=Fpsin(kx).Now consider a “snapshot” of F(x,t) at a later fixed time t.
F
x
v Moving wave
F(x, t) = FP sin (kx - t )
At time zero this is F(x,0)=Fpsin(kx).Now consider a “snapshot” of F(x,t) at a later fixed time t. Then
F(x, t) = FP sin{k[x-(/k)t]}
F
This is the same as the time-zero function, slide to the right a distance (/k)t.
x
v Moving wave
F(x, t) = FP sin (kx - t )
At time zero this is F(x,0)=Fpsin(kx).Now consider a “snapshot” of F(x,t) at a later fixed time t. Then
F(x, t) = FP sin{k[x-(/k)t]}
F
This is the same as the time-zero function, slide to the right a distance (/k)t. The distance it slides to the right changes linearly with time – that is, it moves with a speed v= /k.
The wave moves to the right with speed /k
These are both waves, and both have wave speed /k.
E(x, t) = EP sin (kx-t)
B(x, t) = BP sin (kx-t) z
j
Plane Electromagnetic Waves
These are both waves, and both have wave speed /k.But these expressions for E and B solve Maxwell’s equations only if
E(x, t) = EP sin (kx-t)
B(x, t) = BP sin (kx-t) z
j
€
/k =1/ ε0μ0
Hence the speed of electromagnetic waves is
€
c =1/ ε0μ0 .
Plane Electromagnetic Waves
These are both waves, and both have wave speed /k.But these expressions for E and B solve Maxwell’s equations only if
E(x, t) = EP sin (kx-t)
B(x, t) = BP sin (kx-t) z
j
€
/k =1/ ε0μ0
Hence the speed of electromagnetic waves isMaxwell plugged in the values of the constants and found
€
c =1/ ε0μ0 .
€
c =1/ ε0μ0 = 3 ×108m /s = thespeedof light
Plane Electromagnetic Waves
These are both waves, and both have wave speed /k.But these expressions for E and B solve Maxwell’s equations only if
E(x, t) = EP sin (kx-t)
B(x, t) = BP sin (kx-t) z
j
€
/k =1/ ε0μ0
Hence the speed of electromagnetic waves isMaxwell plugged in the values of the constants and found
€
c =1/ ε0μ0 .
€
c =1/ ε0μ0 = 3 ×108m /s = thespeedof light
Plane Electromagnetic Waves
These are both waves, and both have wave speed /k.But these expressions for E and B solve Maxwell’s equations only if
E(x, t) = EP sin (kx-t)
B(x, t) = BP sin (kx-t) z
j
€
/k =1/ ε0μ0
Hence the speed of electromagnetic waves isMaxwell plugged in the values of the constants and found
€
c =1/ ε0μ0 .
€
c =1/ ε0μ0 = 3 ×108m /s = thespeedof light
Plane Electromagnetic Waves
Thus Maxwell discovered that light is electromagnetic radiation.
Plane Electromagnetic Waves
x
Ey
Bz
• Waves are in phase.• Fields are oriented at 900 to one
another and to the direction of propagation (i.e., are transverse).
• Wave speed is c• At all times E=cB.
€
c =1/ ε0μ0 = 3×108m /s( )
E(x, t) = EP sin (kx-t)
B(x, t) = BP sin (kx-t) z
j
c
The Electromagnetic Spectrum
Radio waves
-wave
infra-red -rays
x-rays
ultra-violet