Maxwell's Equations and Light Waves Vector derivatives: Div, grad, curl, etc. Derivation of wave equation from Maxwell's Equations Why light waves are transverse waves Why we neglect the magnetic field Prof. Rick Trebino Georgia Tech www.physics.gatech.edu/frog/lectures

Div, Grad, Curl, and all that Types of 3D vector derivatives:   The Del operator: The Gradient of a scalar function f : The gradient points in the direction of steepest ascent. If you want to know more about vector calculus, read this book!

Div, Grad, Curl, and all that The Divergence of a vector function: The Divergence is nonzero if there are sources or sinks. A 2D source with a large divergence: x y Note that the x-component of this function changes rapidly in the x direction, etc., the essence of a large divergence.

Div, Grad, Curl, and more all that  The Laplacian of a scalar function :   The Laplacian of a vector function is the same, but for each component of f: The Laplacian tells us the curvature of a vector function.

Div, Grad, Curl, and still more all that Functions that tend to curl around have large curls.

A function with a large curl x y

Lemma: Proof: Look first at the LHS of the above formula:   Taking the 2nd yields: x-component: y-component: z-component:

Lemma (cont’d): Now, look at the RHS:

The equations of optics are Maxwell’s equations. where is the electric field, is the magnetic field, e is the permittivity, and m is the permeability of the medium. As written, they assume no charges (free space).

Derivation of the Wave Equation from Maxwell’s Equations Take of: yielding: Change the order of differentiation on the RHS: But: Substituting for , we have: assuming that m and e are constant in time.

Derivation of the Wave Equation from Maxwell’s Equations (cont’d) Using the lemma,   becomes: But we’ve assumed zero charge density: r = 0, so and we’re left with the Wave Equation! where me = 1/c2

Why light waves are transverse Suppose a wave propagates in the x-direction. Then it’s a function of x and t (and not y or z), so all y- and z-derivatives are zero:      Now, in a charge-free medium, that is, and Substituting the zero values, we have: So the longitudinal fields are at most constant, and not waves.

The magnetic-field direction in a light wave Suppose a wave propagates in the x-direction and has its electric field along the y-direction [so Ex = Ez= 0, and Ey = Ey(x,t)]. What is the direction of the magnetic field?   Use: So: In other words: And the magnetic field points in the z-direction.

The magnetic-field strength in a light wave Suppose a wave propagates in the x-direction and has its electric field in the y-direction. What is the strength of the magnetic field? and Take Bz(x,0) = 0 Differentiating Ey with respect to x yields an ik, and integrating with respect to t yields a 1/-iw. So: But w / k = c:

An Electromagnetic Wave The electric and magnetic fields are in phase. snapshot of the wave at one time The electric field, the magnetic field, and the k-vector are all perpendicular:

The Energy Density of a Light Wave The energy density of an electric field is: The energy density of a magnetic field is: Using B = E/c, and , which together imply that we have: Total energy density: So the electrical and magnetic energy densities in light are equal.

Why we neglect the magnetic field The force on a charge, q, is:   Taking the ratio of the magnitudes of the two forces: Since B = E/c: where is the charge velocity So as long as a charge’s velocity is much less than the speed of light, we can neglect the light’s magnetic force compared to its electric force.

The Poynting Vector: S = c2 e E x B The power per unit area in a beam. Justification (but not a proof): Energy passing through area A in time Dt: = U V = U A c Dt So the energy per unit time per unit area: = U V / ( A Dt ) = U A c Dt / ( A Dt ) = U c = c e E2 = c2 e E B And the direction is reasonable. A c Dt U = Energy density

The Irradiance (often called the Intensity) A light wave’s average power per unit area is the irradiance.   Substituting a light wave into the expression for the Poynting vector, , yields: The average of cos2 is 1/2: real amplitudes

The Irradiance (continued) Since the electric and magnetic fields are perpendicular and B0 = E0 / c, becomes: because the real amplitude squared is the same as the mag-squared complex one. or: where: Remember: this formula only works when the wave is of the form: that is, when all the fields involved have the same

Practical magnitudes for irradiance Sunlight at earth’s surface: 1 kW/m2 Moonlight at earth’s surface: 2 mW/m2 Moon image from http://www.fourmilab.ch/earthview/figures/perigee.jpg Magnitude data and explanation below from http://www.astro.northwestern.edu/labs/m100/mags.html Given that the eye is a logarithmic detector, and the magnitude system is based on the response of the human eye, it follows that the magnitude system is a logarithmic scale. In the original magnitude system, a difference of 5 magnitudes corresponded to a factor of roughly 100 in light intensity. The magnitude system was formalized to assume that a factor of 100 in intensity corresponds exactly to a difference of 5 magnitudes. Since a logarithmic scale is based on multiplicative factors, each magnitude corresponds to a factor of the 5th root of 100, or 2.512, in intensity. The magnitude scale is thus a logarithmic scale in base 1001/5 = 2.512. Object mV Sun -26.8 Full Moon -12.5 Venus -4.4 Jupiter -2.7 Polaris 2.0 Naked-eye limit 6.0 Pluto 15.1 Hubble Telescope 31.0 The magnitude (mv) often used by astronomers is logarithmic. For an increase in mv by five, the irradiance decreases by a factor of 100. So a magnitude difference of 1 corresponds to a factor of 1001/5 or 2.5.

Sums of fields: Electromagnetism is linear, so the principle of Superposition holds. If E1(x,t) and E2(x,t) are solutions to the wave equation, then E1(x,t) + E2(x,t) is also a solution.   Proof: and This means that light beams can pass through each other. It also means that waves can constructively or destructively interfere.

The irradiance of the sum of two waves If they’re both proportional to , then the irradiance is: Different polarizations (say x and y): Intensities add. Same polarizations (say ): Note the cross term! Therefore: The cross term is the origin of interference! Interference only occurs for beams with the same polarization.

The irradiance of the sum of two waves of different color We can’t use the formula because the k’s and w’s are different. So we need to go back to the Poynting vector, This product averages to zero, as does Different colors: Intensities add. Waves of different color (frequency) do not interfere!

Irradiance of a sum of two waves Same polarizations Different polarizations Same colors Different colors Interference only occurs when the waves have the same color and polarization.