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

2. 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

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

4. 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

5. 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

6. 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)

7. 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

8. 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

9. 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

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

11. *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”

12. 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

13. 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

14. 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

15. 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

16. 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!

17. 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

18. 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!

19. 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

20. 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:

21. 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:

22. 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

23. 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

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