Exam 2 Results Max 96.94 Avg 78.16 Std Dev 11.61 Out of 98 points- throwing away MC 3

Right Hand Rule Review I If you curl your fingers in the direction the current flows, thumb points in direction of B-field inside the loop Works for solenoids too B Curl fingers around I and Thumb point to Mag dipole moment. Also gives B inside a loop

Tesla Coils Type of transformer that operates at resonance Uses induction to create high voltages Voltages of approximately 500kV Remember transformers?

AC Circuits AC Sources Often, electrical signals look like sines or cosines AC power, Radio/TV signals, Audio Sine and cosine look nearly identical They are related by a phase shift Cosine wave is advanced by /2 (90 degrees) compared to sine We will always treat source as sine wave Frequency, period, angular frequency related This symbol denotes an arbitrary AC source

House current (US) is at f = 60 Hz and Vrms = 120 V RMS Voltage There are two ways to describe the amplitude Maximum voltage Vmax is an overstatement Average voltage is zero Root-mean-square (RMS) voltage is probably the best way Plot (V)2, find average value, take square root We can do something similar with current House current (US) is at f = 60 Hz and Vrms = 120 V Vmax = 170 V

Amplitude, Frequency, and Phase Shift We will describe any sort of wave in terms of three quantities: The amplitude A is how big it gets To determine it graphically, measure the peak of the wave The frequency is how many times it repeats per second To determine it graphically, measure the period T Frequency f = 1/T and angular frequency  = 2f The phase shift  is how much it is shifted earlier/later compared to basic sine wave Let t0 be when it crosses the origin while rising The phase shift ist0 (radians)

Our Goal Feed AC source through an arbitrary circuit ? Resistors, capacitors, inductors, or combinations of them We will always assume the incoming wave has zero phase shift ? We want to find current as a function of time For these components, can show angular frequency  is the same We still need to find amplitude Imax and phase shift  for current Also want instantaneous power P and average powerP consumed Generally, maximum current will be proportional to maximum voltage Call the ratio the impedance, Z

Degrees vs. Radians All my calculations will be done in radians Degrees are very commonly used as well But the formulas look different Probably best to set your calculator on radians and leave it there

Resistors R = 1.4 k Can find the current from Ohm’s Law The current is in phase with the voltage Voltage Current Impedance vector: A vector showing relationship between voltage and current Length, R is the ratio Direction is to the right, representing the phase shift of zero 1.4 k

Power in Resistors R = 1.4 k We want to know Instantaneous power Average Power

*We will ignore the term “reactance” Capacitors C= 2.0 F Charge of capacitor is proportional to voltage Current is derivative of charge Current leads voltage by /2 We say there is a –/2 phase shift: Impedance vector: Define the impedance* for a capacitor as: Make a vector out of it Length XC Pointing down for  = –½ 1.3 k *We will ignore the term “reactance”

Only resistors contribute to the average power P consumed Power in Capacitors C= 2.0 F We want to know Instantaneous Power Average Power Power flows into and out of capacitor No net power is consumed by capacitor Only resistors contribute to the average power P consumed

Capacitors and Resistors Combined Capacitors and resistors both limit the current – they both have impedance Resistors: same impedance at all frequencies Capacitors: more impedance at low frequencies

Concept Question L C R f Vmax How will XL and XC compare at the frequency where the maximum power is delivered to the resistor? A) XL > XC B) XL < XC C) XL = XC D) Insufficient information Resonance happens when XL = XC. This makes Z the smallest It happens only at one frequency Same frequency we got for LC circuit

Loop has unin-tended inductance Concept Question The circuit at right is in a steady state. What will the voltmeter read as soon as the switch is opened? A) 0.l V B) 1 V C) 10 V D) 100 V E) 1000 V R1 = 10  L I = 1 A V R2 = 1 k + – E = 10 V The current remains constant at 1 A It must pass through resistor R2 The voltage is given by V = IR Note that inductors can produce very high voltages Inductance causes sparks to jump when you turn a switch off + – Loop has unin-tended inductance

Maxwell’s Equations Not quite complete! We now have four formulas that describe how to get electric and magnetic fields from charges and currents Gauss’s Law Gauss’s Law for Magnetism Ampere’s Law Faraday’s Law There is also a formula for forces on charges Called Lorentz Force One of these is wrong!

Ampere’s Law is Wrong! Maxwell realized Ampere’s Law is not self-consistent This isn’t an experimental argument, but a theoretical one Consider a parallel plate capacitor getting charged by a wire Consider an Ampere surface between the plates Consider an Ampere surface in front of plates But they must give the same answer! There must be something else that creates B-fields Note that for the first surface, there is also an electric field accumulating in capacitor Maybe electric fields? Take the time derivative of this formula Speculate : This replaces I for first surface I I

Ampere’s Law (New Recipe) Is this self-consistent? Consider two surfaces with the same boundary I1 I2 Gauss’s Law for electric fields: E2 E1 B This makes sense!

Maxwell’s Equations This is not the form in which Maxwell’s Equations are usually written It involves complicated integrals It involves long-range effects Our first goal – rewrite them as local equations Make the volumes very small Make the loops very small Large volumes and loops can be made from small ones If it works on the small scale, it will work on the large Skip slides

Gauss’s Law for Small Volumes (2) Consider a cube of side a One corner at point (x,y,z) a will be assumed to be very small Gauss’s Law says: Let’s get flux on front and back face: x y z a Now include the other four faces:

Gauss’s Law for Small Volumes Divide both sides by a3, the volume q/V is called charge density  A similar computation works for Gauss’s Law for magnetic fields: A more mathematically sophisticated notation allows you to write these more succinctly:

Ampere’s Law for Small Loops Consider a square loop a One corner at point (x,y,z) a will be assumed to be very small Ampere’s Law says: Let’s get integral on top and bottom z a y x a Add the left and right sides Calculate the electric flux Put it together

Ampere’s Law for Small Loops (2) z a y x Divide by a2 Current density J is I/A Only in x-direction counts Redo it for loops oriented in the other two directions Similar formulas can be found for Faraday’s Law a

Maxwell’s Equations: Differential Form In more sophisticated notation: