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 : two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows.

The Divergence of a vector function: The Divergence is nonzero if there are sources or sinks. The divergence of a vector field is relatively easy to understand intuitively. Imagine that the vector field F below gives the velocity of some fluid flow. It appears that the fluid is exploding outward from the origin. In contrast, the below vector field represents fluid flowing so that it compresses as it moves toward the origin. Since this compression of fluid is the opposite of expansion, the divergence of this vector field is negative.

 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 x y

Home work: Proof

2-2. Waves and Wave Equation

A wave is anything that moves. f(x) f(x-3) f(x-2) f(x-1) x 0 1 2 3 What is waves? 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. 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.

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: 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:

Now let's consider the electric wave We use cosine- and sine-wave solutions: 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 + i y = A cos() + i A sin() where i = (-1)1/2

Euler's Formula exp(ij) = cos(j) + i sin(j) so the point, P = A cos(j) + i A sin(j), can be written: P = A exp(ij) where A = Amplitude, j = 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(ij): A2 = Re{ z }2 + Im{ z }2 tan(j) = 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(ij) = cos(j) + i sin(j), 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 1D to 3D which has the solution: where k

2-3. Maxwell's Equations Gauss's law =0 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: r = 0, so and we’re left with the Wave Equation! where Similarly,

Light wave is transverse [횡파] (1) Motion is along the direction of propagation—longitudinal polarization Longitudinal: Motion is transverse to the direction of propagation—transverse polarization Transverse: 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)