The equations so far Gauss Law for E

The equations so far. . . Gauss’ Law for E Fields Gauss’ Law for B Fields Faraday’s Law Ampere’s Law 1 2/26/2021

Ampere’s Law No current inside Current inside

Maxwell’s Displacement Current, Id 3 2/26/2021

Maxwell’s Approach Time varying magnetic field leads to curly electric field. Time varying electric field leads to curly magnetic field? I ‘equivalent’ current combine with current in Ampere’s law 4

Maxwell’s Equations (1865) in Systeme International (SI or mks) units 5 2/26/2021

Question Suppose you were able to charge a capacitor with constant current (does not change in time). Does a B field exist in between the plates of the capacitor? A) NO B) YES 6 2/26/2021

Maxwell’s Equations (Free Space) Note the symmetry of Maxwell’s Equations in free space, when no charges or currents are present We can predict the existence of electromagnetic waves. Why? Because the wave equation is contained in these equations. Remember the wave equation. h is the variable that is changing in space (x) and time (t). v is the velocity of the wave. 7 2/26/2021

Review of Waves from Mechanics The one-dimensional wave equation: has a general solution of the form: A solution for waves traveling in the +x direction is: 2/26/2021 8

Wave Examples Wave on a String: Electromagnetic Wave e. g. , sqrt(tension/mass) is wave speed of a guitar string, proportional to frequency of fundamental What is waving? ? The Electric & Magnetic Fields !! Rewrite Maxwell’s equations as equations of the form: The velocity of the wave, v, will be related to 0 and 0. 2/26/2021 9

Four Step Plane Wave Derivation Step 1 Assume we have a plane wave propagating in z (i. e. E, B not functions of x or y) Example: Step 2 Apply Faraday’s Law to infinitesimal loop in x-z plane x Ex Ex z 1 y By z 2 x z Z 10 2/26/2021

Four Step Plane Wave Derivation Step 3 Apply Ampere’s Law to an infinitesimal loop in the y-z plane: x Ex Z z 1 z 2 y By By z y Step 4: Use results from steps 2 and 3 to eliminate By 11 2/26/2021

Velocity of Electromagnetic Waves We derived the wave equation for Ex: The velocity of electromagnetic waves in free space is: Putting in the measured values for 0 & 0, we get: This value is identical to the measured speed of light! We identify light as an electromagnetic wave. 2/26/2021 12

Maxwell Equations: Electromagnetic Waves Maxwell’s Equations contain the wave equation The velocity of electromagnetic waves: c = 2. 99792458 x 108 m/s The relationship between E and B in an EM wave Energy in EM waves: the Poynting vector x z y 2/26/2021 13

Question If the magnetic field of a light wave oscillates parallel to a y axis and is given by By = Bm sin(kz- t) in what direction does the wave travel? A. -y B. -z C. y D. z E. -x 14 2/26/2021

Question If the magnetic field of a light wave oscillates parallel to a y axis and is given by By = Bm sin(kz+ t) in what direction does the wave travel and parallel to which axis does the associated electric field oscillate? A. -z, y B. z, x C. -z, x D. z, -x E. -z, -x 2/26/2021 15

Electromagnetic Spectrum ~1850: infrared, visible, and ultraviolet light were the only forms of electromagnetic waves known. Visible light (human eye) 16 2/26/2021

Electromagnetic Spectrum 17 2/26/2021

Wien’s Displacement Law 2/26/2021

White Light: A Mixture of Colors (DEMO) Demos: 7 C-1 7 C-2 2/26/2021

Spectral Lines Energy states of an atom are discrete and so are the energy transitions that cause the emission of a pho (DEMO) 2/26/2021

How is B related to E? We derived the wave equation for Ex: We could have derived for By: How are Ex and By related in phase and magnitude? Consider the harmonic solution: where 25 2/26/2021

E & B in Electromagnetic Waves Plane Wave: where: x z y The direction of propagation is given by the cross product where are the unit vectors in the (E, B) directions. Nothing special about (Ex, By); eg could have (Ey, -Bx) Note cyclical relation: 2/26/2021 26

Energy in Electromagnetic Waves Electromagnetic waves contain energy. We know the energy density stored in E and B fields: In an EM wave, B = E/c The total energy density in an EM wave = u, where The Intensity of a wave is defined as the average power (Pav=uav/ t) transmitted per unit area = average energy density times wave velocity: • For ease in calculation define Z 0 as: 2/26/2021 27

The Poynting Vector The direction of the propagation of the electromagnetic wave is given by: This energy transport is defined by the Poynting vector S as: S has the direction of propagation of the wave The magnitude of S is directly related to the energy being transported by the wave The intensity for harmonic waves is then given by: 28 2/26/2021

Characteristics x z S 29 2/26/2021

Summary of Electromagnetic Radiation combined Faraday’s Law and Ampere’s Law – time varying B-field induces E-field – time varying E-field induces B-field • E-field and B-field are perpendicular • energy density • • Poynting Vector describes power flow • units: watts/m 2 E S B 30 2/26/2021