Exam is Tuesday Nov. 25th in class Thursday, November 12, 2003 WARNING: Exam is Tuesday Nov. 25th in class Review Sessions: In class: Friday 12-1 pm; 2-3 pm Me: Monday 4-6 pm 35-225 Office Hours: Brian F.: Friday 1-2 pm 4-344 Various: Sunday 1-5 pm TEAL Brian P.: Monday 3-4 pm 4-344 Monday 6-8 pm 6-106 Please email me questions Class 28

Class 30: Outline Hour 1: Displacement Current Hour 2: Thursday, November 12, 2003 Class 30: Outline Hour 1: Displacement Current Hour 2: Electromagnetic waves Class 28

Last Time: Driven RLC Circuits

Resistance Reactance Impedance AC Circuits: Summary Element I0 Current vs. Voltage Resistance Reactance Impedance Resistor In Phase Capacitor Leads Inductor Lags Although derived from single element circuits, these relationships hold generally!

Driven RLC Series Circuit Impedance

Resonance C-like: f < 0 I leads L-like: f > 0 I lags

This Time: Putting it All Together

Displacement Current

Ampere’s Law: Capacitor Consider a charging capacitor: I Use Ampere’s Law to calculate the magnetic field just above the top plate 1) Red Amperian Area, Ienc= I 2) Green Amperian Area, I = 0 What’s Going On?

This is called (for historic reasons) the Displacement Current We don’t have current between the capacitor plates but we do have a changing E field. Can we “make” a current out of that? This is called (for historic reasons) the Displacement Current

Maxwell-Ampere’s Law

PRS Questions: Capacitor

In Class Problem: Displacement Current

Maxwell’s Equations

Electromagnetism Review E fields are created by: (1) electric charges (2) time changing B fields B fields are created by (1) moving electric charges (NOT magnetic charges) (2) time changing E fields E (B) fields exert forces on (moving) electric charges Gauss’s Law Faraday’s Law Ampere’s Law Maxwell’s Addition Lorentz Force

Maxwell’s Equations

Electromagnetic Radiation

A Question of Time…

Electromagnetic Radiation: Plane Waves

Traveling Waves Consider f(x) = x=0 What is g(x,t) = f(x-vt)? x=0 t=0 t=t0 x=2vt0 t=2t0 f(x-vt) is traveling wave moving to the right!

Traveling Sine Wave Now consider f(x) = y = y0sin(kx): Amplitude (y0) What is g(x,t) = f(x+vt)? Travels to left at velocity v y = y0sin(k(x+vt)) = y0sin(kx+kvt)

Traveling Sine Wave At x=0, just a function of time: Amplitude (y0)

Traveling Sine Wave

Electromagnetic Waves Hz Remember:

Electromagnetic Radiation: Plane Waves Watch 2 Ways: 1) Sine wave traveling to right (+x) 2) Collection of out of phase oscillators (watch one position) Don’t confuse vectors with heights – they are magnitudes of E (gold) and B (blue)

PRS Question: Wave

Group Work: Java Problem 1

Properties of EM Waves Travel (through vacuum) with speed of light At every point in the wave and any instant of time, E and B are in phase with one another, with E and B fields perpendicular to one another, and to the direction of propagation (they are transverse):

Direction of Propagation

PRS Question: Direction of Propagation

Energy & the Poynting Vector

Energy in EM Waves Energy densities: Consider cylinder: What is rate of energy flow per unit area?

Poynting Vector and Intensity Direction of energy flow = direction of wave propagation units: Joules per square meter per sec Intensity I:

Energy Flow: Resistor On surface of resistor is INWARD

PRS Questions: Poynting Vector

Energy Flow: Inductor On surface of inductor with increasing current is INWARD

Energy Flow: Inductor On surface of inductor with decreasing current is OUTWARD

In Class Problem: Poynting Vector

Momentum & Radiation Pressure EM waves transport energy: They also transport momentum: And exert a pressure: This is only for hitting an absorbing surface. For hitting a perfectly reflecting surface the values are doubled:

Standing Waves

Standing Waves What happens if two waves headed in opposite directions are allowed to interfere?

Standing Waves: Who Cares? Most commonly seen in resonating systems: Musical Instruments, Microwave Ovens

Standing Waves: Bridge

Microwave Ovens: Hot Spots Can you measure the speed of light with marshmallows?

Microwave Ovens: Hot Spots

Microwave Ovens: Hot Spots OR

Microwave Ovens: Hot Spots

Microwave Ovens: Hot Spots

Group Work: Standing Waves Play with Problem 2 in Java Example!