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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