Electricity and Magnetism Physics 208 Dr. Tatiana Erukhimova Waves and Review

Coulomb’s Law Conservation of electric charge is the permittivity of free space Conservation of electric charge Charge Charge is conserved: in any isolated system, the total charge cannot change. If it does change, then the system is not isolated: charge either went somewhere or came in from somewhere

Like charges repel; opposites attract Let’s denote the force that exerts on as and force exerted by on as ; r is the distance between charges. (Newton’s third law works!) Like charges repel; opposites attract

Exercise: If two electrons are placed meters apart, what is the magnitude of the Coulomb force between them? Compare this to the gravitational force between them. r Solution: The magnitude of electric force The magnitude of gravitational force (no matter what the separation is)

Gauss’s Law A conducting sphere, conducting shell, insulating sphere, shell …..

Two parallel conducting plates - + a d (the total field at any point between the plates)

Capacitors Consider two large metal plates which are parallel to each other and separated by a distance small compared with their width. Area A L The field between plates is

The capacitance is:

Capacitors in series: Capacitors in parallel:

Current, Ohm’s Law, Etc.

Current Density Consider current flowing in a homogeneous wire with cross sectional area A.

Circuits will be included! For steady state situation Circuits will be included!

Joule’s Law

The force on a charge q moving with a velocity If the magnitude of the force

The angular velocity

Using Crossed and Fields Velocity selector independent of the mass of the particle!

Ampere’s Law The field produced by an infinite wire

Biot-Savart Law Infinitesimally small element of a current carrying wire produces an infinitesimally small magnetic field is called permeability of free space (Also called Ampere’s principle)

Force exerted on a current carrying wire For a straight, finite wire of length and uniform magnetic field

Faraday’s Law of Induction The induced EMF in a closed loop equals the negative of the time rate of change of magnetic flux through the loop

There can be EMF produced in a number of ways: A time varying magnetic field An area whose size is varying A time varying angle between and Any combination of the above

From Faraday’s law: a time varying flux through a circuit will induce an EMF in the circuit. If the circuit consists only of a loop of wire with one resistor, with resistance R, a current R Which way? Lenz’s Law: if a current is induced by some change, the direction of the current is such that it opposes the change.

A Simple Generator

Faraday’s Law is used to obtain differential equations for some simple circuits. Self-inductance L

Displacement current

No charges, no real currents Wave equation

k is a wave number,  is a wave length, T is the period Velocity of propagation

Waves can be represented by simple harmonic motion

Standing wave y = Asin(kx − ωt) + Asin(kx + ωt)

The amplitude of a wave is a measure of the maximum disturbance in the medium during one wave cycle. (the maximum distance from the highest point of the crest to the equilibrium). The wavelength (denoted as λ) is the distance between two sequential crests (or troughs). This generally has the unit of meters. A wavenumber Phase velocity: Period

Electromagnetic waves

Light as a Wave (1) f = c/l l c = 300,000 km/s = 3*108 m/s Light waves are characterized by a wavelength l and a frequency f. f and l are related through f = c/l

The Electromagnetic Spectrum Wavelength Frequency High flying air planes or satellites Need satellites to observe

Dual, wave-particle nature of light 1 eV = 1.6x10-19 J c = 3x108 m/s 1 Angstrom = 10-10 m Speed of light in matter: cm = c/n, where n is refractive index Note: n is a function of 

Light as a Wave (2) Wavelengths of light are measured in units of nanometers (nm) or Ångström (Å): 1 nm = 10-9 m 1 Å = 10-10 m = 0.1 nm Visible light has wavelengths between 4000 Å and 7000 Å (= 400 – 700 nm).

Light as Particles E = h*f Light can also appear as particles, called photons (explains, e.g., photoelectric effect). A photon has a specific energy E, proportional to the frequency f: E = h*f h = 6.626x10-34 J*s is the Planck constant. The energy of a photon does not depend on the intensity of the light!!!

Maxwell’s Equations

Information Age The cost of the transmission, storage and processing of data has been decreasing extremely fast Information is available anytime, any place, and for everyone Information and knowledge became a capital asset All of this became possible because of several revolutionary ideas

Telecommunications Key part of information revolution is telecommunications – ability to send information over long distance

How it all started … Samuel Morse's telegraph key, 1844. Today's information age began with the telegraph. It was the first instrument to transform information into electrical signal and transmit it reliably over long distances. Alexander Graham Bell’s commercial telephone from 1877.

Speaking into the handset's transmitter or microphone makes its diaphragm vibrate. This varies the electric current, causing the receiver's diaphragm to vibrate. This duplicates the original sound. Telephone connection requires a dedicated wire line; Only one communication is possible at a time

Radio: communication through radio waves 1895 www.nrao.edu Frequency measured in Hertz 1 Hz = 1 cycle/second 1 kHz = 1000 cycles/second Alexander Popov Guglielmo Marconi How many channels are possible? How many signals can be transmitted at the same time??

Radio stations have to broadcast at different carrier frequencies to avoid cross-talk Human ear: 10 Hz-20 kHz Range of frequencies (Bandwidth) needs to be at least 20 kHz for each station Frequencies of different stations should be at least 20 kHz apart

Need more channels? Need higher speed? Use higher frequencies for transmission! Higher carrier frequencies Wider bandwidth Higher data rate, more channels Using light? Optical frequencies ~ 1014 Hz !

Sand?! How can we send light over long distances? Air? Only within line of sight; High absorption and scattering, especially when it rains Are there any “light wires” (optical waveguides)? Copper wire? High absorption, narrow bandwidth 300 MHz Glass? Window glass absorbs 90% of light after 1 m. Only 1% transmission after 2 meters. Sand?!

Ultra-low absorption in silica glasses Transmisson 95.5% of power after 1 km 1% of power after 100 km: need amplifiers and repeaters Total bandwidth ~ 100,000 GHz!! Predicted 1965, in first low-loss fiber in 1970 Silica (Silicon dioxide) is sand – the most abundant mineral on Earth

How to trap light with transparent material?? Total internal reflection! n1 > n2 Light coming from more refractive to less refractive medium experiences total reflection – get trapped there!

This is a photograph of the pale blue dot, Earth, taken by the Voyager 1 spacecraft at a distance of 3.7 billion kilometers away. The spacecraft had completed its primary mission and was passing Saturn, hurtling through space at 40,000 mph. Carl Sagan requested that the spacecraft turn around and take a photo of Earth, not for any scientific purpose, but as a sobering reminder of our planet's insignificance. The Pale Blue Dot

https://www.youtube.com/watch?v=wupToqz1e2g&feature=youtu.be

Thank you for a great semester!