Maxwell’s equations(sec. 32.1) Plane EM waves & speed of light(sec. 32.2) The EM spectrum(sec. 32.6) Electromagnetic Waves Ch. 32 C 2012 J. F. Becker

Learning Goals - we will learn: ch 32 Maxwell’s Equations – the four fundamental equations of EM theory. How the speed of light is related to the fundamental constants of electricity and magnetism (   and   ). How to describe propagating and standing EM waves.

MAXWELL’S EQUATIONS C 2004 Pearson Educational / Addison Wesley The relationships between electric and magnetic fields and their sources can be stated compactly in four equations, called Maxwell’s equations. Together they form a complete basis for the relation of E and B fields to their sources.

A capacitor being charged by a current i c has a displacement current equal to i C between the plates, with displacement current i D = e A dE/dt. This changing E field can be regarded as the source of the magnetic field between the plates.

A capacitor being charged by a current i C has a displacement current equal to i C in magnitude between the plates, with DISPLACEMENT CURRENT i D = e A dE/dt From C = e A / d and D V = E d we can use q = C V to get q = ( e A / d ) (E d ) = e E A = e F E and from i C = dq / dt = e A dE / dt = e d F E / dt = i D We have now seen that a changing E field can produce a B field and from Faraday’s Law a changing B field can produce an E field (or emf) C 2012 J. Becker

MAXWELL’S EQUATIONS C 2004 Pearson Educational / Addison Wesley The relationships between electric and magnetic fields and their sources can be stated compactly in four equations, called Maxwell’s equations. Together they form a complete basis for the relation of E and B fields to their sources.

An electromagnetic wave front. The plane representing the wave front (yellow) moves to the right with speed c. The E and B fields are uniform over the region behind the wave front but are zero everywhere in front of it.

Gaussian surface for an electromagnetic wave propagating through empty space. The total electric flux and total magnetic flux through the surface are both zero. Both E and B are _ to the direction of propagation.

Applying Faraday’s Law to a plane wave.  E dl = -d/dt{  B }= - d/dt  B dA LH:  E o dl = -Ea RH: In time dt the wave front moves to the right a distance c dt. The magnetic flux through the rectangle in the xy-plane increases by an amount d  B equal to the flux through the shaded rectangle in the xy-plane with area ac dt, that is, d  B =B ac dt; d  B /dt = B ac -d  B / dt = -B ac and (LH = RH): -Ea = -B ac. So E = Bc

Applying Ampere’s Law to a plane wave: i C = 0 LH:  B o dl = Ba RH: In time dt the wave front moves to the right a distance c dt. The electric flux through the rectangle in the xz-plane increases by an amount d  E equal to E times the area ac dt of the shaded rectangle, that is, d  E = E ac dt. Thus d  E / dt = E ac, and (LH = RH): Ba =     Eac  B =     Ec and from E = Bc and B =     Ec we must have c = 1 / (     ) 1/2  B dl =   i C +      d  E /dt = 3.00 (  ) 8 m/sec

Faraday’s Law applied to a rectangle with height a and width  x parallel to the xy-plane.

Ampere’s Law applied to a rectangle with height a and width  x parallel to the xz-plane.

Representation of the electric and magnetic fields in a propagating wave. One wavelength is shown at time t = 0. Propagation direction is E x B. WAVE PROPAGATION SPEED c = 1 / (     ) 1/2 c = 3.00 (  ) 8 m/sec

Wave front at time dt after it passes through a stationary plane with area A. The volume between the plane and the wave front contains an amount of electromagnetic energy uAc dt.

ENERGY AND MONENTUM IN EM WAVES Energy density: u =   E 2 /2 + B 2 /2   (Ch 30) Using B = E/c = E (     ) 1/2 we get u =   E 2 /2 + E 2 (     ) /2   u =   E 2 /2 +   E 2 /2 =   E 2 u =   E 2 (half in E and half in B) (eqn 32.25)

ENERGY FLOW IN EM WAVES dU = u dV =   E 2 (Ac dt) Define the “Poynting” vector S = energy flow / time x area S = dU / dt A =   E 2 (Ac) / A =    c E 2 or S =   c E 2 =   E 2 / (     ) 1/2 = (   /   ) 1/2  E 2 = EB /   And define the “Poynting” vector: S = E x B /    With units of Joule/sec meter 2 or Watt/meter 2

Wave front at time dt after it passes through a stationary plane with area A. The volume between the plane and the wave front contains an amount of electromagnetic energy uAc dt.

A standing electromagnetic wave does not propagate along the x-axis; instead, at every point on the x-axis the E and B fields simply oscillate.

Examples of standing electromagnetic waves Microwave ovens have a standing wave with l = 12.2 cm, a wavelength that is strongly absorbed by water in foods. Because the wave has nodes (zeros) every 6.1 cm the food must be rotated with cooking to avoid cold spots! Lasers are made of cavities of length L with highly reflecting mirrors at each end to reflect waves with wavelengths that satisfy L = m l / 2, where m = 1, 2, 3, … C 2012 J. Becker

THE ELECTROMAGNETIC SPECTRUM The frequencies and wavelengths found in nature extend over a wide range. The visible wavelengths extend from approximately 400 nm (blue) to 700 nm (red).

One cycle in the production of an electro-magnetic wave by an oscillating electric dipole antenna. The red arrows represent the E field. (B not shown.)

PREPARATION FOR FINAL EXAM At a minimum the following should be reviewed: Gauss's Law - calculation of the magnitude of the electric field caused by continuous distributions of charge starting with Gauss's Law and completing all the steps including evaluation of the integrals. Ampere's Law - calculation of the magnitude of the magnetic field caused by electric currents using Ampere's Law (all steps including evaluation of the integrals). Faraday's Law and Lenz's Law - calculation of induced voltage and current, including the direction of the induced current. Calculation of integrals to obtain values of electric field, electric potential, and magnetic field caused by continuous distributions of electric charge and current configurations (includes the Law of Biot and Savart for magnetic fields). Maxwell's equations - Maxwell's contribution and significance. DC circuits - Ohm's Law, Kirchhoff's Rules, series-parallel R’s, RC ckts, power. Series RLC circuits - phasors, phase angle, current, power factor, average power. Vectors - as used throughout the entire course.

See Review C 2012 J. F. Becker