The equations so far..... Gauss’ Law for E Fields Gauss’ Law for B Fields Faraday’s Law Ampere’s Law 11/15/2018

Ampere’s Law Current inside No current inside Lets work with the last one… Current inside No current inside

Maxwell’s Displacement Current, Id 11/15/2018

Maxwell’s Approach Time varying magnetic field leads to curly electric field. Time varying electric field leads to curly magnetic field? Current in wire I – causes change in E flux, should cause the same effect in curly B I ‘equivalent’ current combine with current in Ampere’s law

Maxwell’s Equations (1865) in Systeme International (SI or mks) units 11/15/2018

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 11/15/2018

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. 11/15/2018

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: 11/15/2018

Wave Examples Electromagnetic Wave Wave on a String: e.g., sqrt(tension/mass) is wave speed of a guitar string, proportional to frequency of fundamental Electromagnetic Wave 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. 11/15/2018

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 x z1 z2 z By Z y 11/15/2018

Four Step Plane Wave Derivation Step 3 Apply Ampere’s Law to an infinitesimal loop in the y-z plane: x z y z1 z2 By Z y Ex Step 4: Use results from steps 2 and 3 to eliminate By 11/15/2018

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. 11/15/2018

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 11/15/2018

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? -y -z y z -x 11/15/2018

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? -z, y z, x -z, x z, -x -z, -x 11/15/2018

Electromagnetic Spectrum ~1850: infrared, visible, and ultraviolet light were the only forms of electromagnetic waves known. Visible light (human eye) 11/15/2018

Electro- magnetic Spectrum 11/15/2018

Wien’s Displacement Law 11/15/2018

White Light: A Mixture of Colors (DEMO) Demos: 7C-1 7C-2 11/15/2018

Spectral Lines Energy states of an atom are discrete and so are the energy transitions that cause the emission of a photon (DEMO) 11/15/2018

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 11/15/2018

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: 11/15/2018

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 Z0 as: 11/15/2018

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: 11/15/2018

Characteristics x S z 11/15/2018

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/m2 E B S 11/15/2018