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2. Maxwell's Equations and Light Waves
2-1. Math (div, grad, curl or rot)
Types of 3D differential vector operator, del
The gradient of a scalar function f :
The Divergence of a vector function:
The Divergence is nonzero if there are sources or sinks.
The Laplacian of a scalar function :
The Laplacian of a vector function is the same,
but for each component of g:
The curl of a vector function:
The curl can be treated as a matrix determinant:
Example: Calculate
f (x, y, z) = (-y, x, 0)
f (1, 0, 0) = (0,1, 0)
f (0,1, 0) = (-1, 0, 0)
f (-1, 0, 0) = (0, -1, 0)
f (0, -1, 0) = (1, 0, 0)
y
x
Home work: Proof
2-2. Waves and Wave Equation
What is waves?
f(x)
f(x-2)
f(x-1)
f(x-3)
A wave is anything that moves.
To displace any function f(x) to the right,
just change its argument from x to x-a,
where a is a positive number.
0
1
2
3
If we let a = v t, where v is positive and t is time, then the
displacement will increase with time.
So f(x-vt) represents a rightward, or forward, propagating wave.
Similarly, f(x+vt) represents a leftward, or backward, propagating
wave.
v will be the velocity of the wave.
x
The 1D wave equation and its solution
We’ll derive the wave equation from Maxwell’s equations later.
Here it is in its one-dimensional form for scalar functions, f:
¶2 f 1 ¶2 f
- 2 2 =0
2
¶x v ¶t
Light waves (or electromagnetic wave) will be a solution to this equation. And
v will be the velocity of light.
The wave equation has the simple solution:
f ( x,t ) = f ( x ± vt )
Now let's consider the electric wave
¶ 2E
¶ 2E
- me 2 = 0
2
¶x
¶t
¶2 f 1 ¶2 f
- 2 2 =0
2
¶x v ¶t
We use cosine- and sine-wave solutions:
E ( x, t ) = B cos[k ( x ± vt )] + C sin[k ( x ± vt )]
kx ± (kv)t
E ( x, t ) = B cos(kx ± wt ) + C sin(kx ± wt )
w
k
=
1
me
=v Þc
where : permittivity of free space, : permeability of free space.
E(x,t) = B cos(kx – wt) + C sin(kx – wt)
= A cos(kx – wt – q)
Spatial quantities:
Temporal quantities:
Complex numbers
Consider a point, P = (x,y), on a 2D
Cartesian grid.
Instead of using an ordered pair, (x,y), we write:
P = x+iy
= A cos() + i A sin()
where i = (-1)1/2
Euler's Formula
exp(i) = cos() + i sin()
so the point, P = A cos() + i A sin(), can be written:
P = A exp(i)
where
A = Amplitude,  = Phase
Any complex number, z, can be written:
z = Re{ z } + i Im{ z }
So
Re{ z } = 1/2 ( z + z* )
and
Im{ z } = 1/2i ( z – z* )
where z* is the complex conjugate of z ( i  –i )
The "magnitude," | z |, of a complex number is:
| z |2 = z z* = Re{ z }2 + Im{ z }2
To convert z into polar form, A exp(i):
A2 = Re{ z }2 + Im{ z }2
tan() = Im{ z } / Re{ z }
Waves using complex numbers
The electric field of a light wave
E(x,t) = A cos(kx – wt – q)
can be expressed by using complex numbers.
Since exp(i) = cos() + i sin(), E(x,t) can be written:
E(x,t) = Re { A exp[i(kx – wt – q)] }
We often leave out 'Re'.
The 3D wave equation
¶ 2E
¶ 2E
- me 2 = 0
2
¶x
¶t
1D to 3D
which has the solution: E( x, y,z,t ) = E 0 exp[i(k × r - wt )]
where
k = ( k1, k2 , k3 )
r = ( r1, r2, r3 )
k
2-3. Maxwell's Equations
=0
Gauss's law
Gauss's law for magnetism
Maxwell-Faraday equation
Ampere's circuital law
• E: electric field, B: magnetic field, : permittivity of free space, : permeability of free space
• We assumed =0.
Derivation of the Wave Equation
from Maxwell’s Equations (1)
Take curl of 3rd Eq
yielding:
Change the order of differentiation on the right hand side:
But from 4th Eq.:
Substituting for
, we have:
Assuming that  and  are constant in time, we have
Using a theformula
becomes:
But we’ve assumed zero charge density:  = 0, so
and we’re left with the Wave Equation!
where
Similarly,
c= 1
em
Light wave is transverse [횡파] (1)
Longitudinal:
Transverse:
Motion
is along
the
direction of propagation—
longitudinal polarization
Motion is transverse to the
direction of propagation—
transverse polarization
Space has 3 dimensions, of which 2 are transverse to the propagation
direction, so there are 2 transverse waves in addition to the potential
longitudinal one.
The direction of the wave’s variations is called its polarization.
Light wave is transverse [횡파] (2)
c= 1
em