*Updated: 13 November 2015 Friday.*

Monday 11/9: **LC Oscillator circuit.** Same 2nd order differential
equation as the Simple Harmonic Oscillator (PHYS-2050), such as a mass on
a spring. Solutions are sines and cosines. Energy is held constant for all
*t* between the capacitor and the inductor. Can't really have a true
LC oscillator, since normal wires and coils have a resistance which
dissipates energy through Joule heating. LC Oscillator solution: q(t) = Q_{0}
cos(omega t + phi), where *omega = 1 / SQRT (LC)* is the angular
frequency and *phi* is a phase angle. Energy is held constant for
all *t* between the capacitor and the inductor. U = U_{C }+
U_{L }= q²/2C + ½Li² = Q²/2C = ½ L I ².
Can't really have a true LC oscillator, since normal wires and coils have
a resistance which dissipates energy through Joule heating. **Comments
ONLY about the RLC Damped Harmonic Oscillator**. Mechanical analogue is
the mass-on-a-spring with shock absorbers.**A.C. Circuits**. Voltage
is a sine or cosine functions, as is the Current. Problem: Average voltage
is ZERO. Need to define a new average, the Root-Mean-Square. It is the
RMS Voltage and Current that are usually
reported in A.C. circuits. Typical A.C. frequency in U.S. is 60 Hz. Need
to specify what type of A.C. For sine wave, define RMS Voltage as 0.7071
Maximum Voltage. Similar for RMS Current.

- Note: In class today I think I used the form U
_{C }= ½Cv², rather than q²/2C.

Tuesday 11/10: **Mutual Inductance
between two inductors**. 2nd coil responds only to changes in magnetic
flux coming from 1st coil, which is based on the changes in the current
*i _{1}* in the 1st coil. And vice versa.

Wednesday 11/11: **Homework:** Repeat the power loss and efficiency
work we did Tuesday, but since we're using AC, we can change the voltage
at a transformer just outside to a higher voltage. This changes the
current I, the power loss due to Joule heating in the powerlines (P = I²R)
and the efficiency (Useful Power / Total Power). Remember that the Useful
power we are planning to deliver is 50,000 W, or 466.7A at 120 V_{rms}.
Change the voltage to 1200 volts, 12,000 volts, 120,000 volts and
1,200,000 volts. Q10 in-class.

- Note that degrees and radians are what Dr. Phil calls "quasi-units"
-- they aren't really units because we can make this appear or disappear
as needed. omega, the angular frequency, is in rad/sec. f, the
frequency, is in Hz = 1/sec = sec
^{-1}. - NOTE: Series RL current has been fixed. (11-11-2015 We) This correct did NOT affect the solution to Q10.

Thursday 11/12: and Friday 11/13: Dr. Phil sick -- substitutes. Available material includes:

Why A.C. power? (2) D.C. power lines have huge power losses due to Joule
heating, very low efficiency. Actual Efficiency = Power Used ÷ Total
Power Generated. Power lines run at higher voltages to minimize power
losses due to Joule heating in the powerlines (P = I²R). For
resistive only circuits, can still use Ohm's Law, V = I R. Current and
Voltage are both sine waves. Real A.C. circuits may have a Resistive
nature, a Capacitive nature and an Inductive nature. For A.C. circuits
with a Resistor only: I and V stay in phase with each other. RL Circuits:
I and V out of phase by -90°. (The current *lags* behind the
voltage.) Inductive Reactance. Phasor
diagrams -- taking the y-component of a rotating vector gives the sine
function. RC Circuits: I and V out of phase by +90°. (The current
*leads* ahead of the voltage.) Capacitive
Reactance. Many A.C. circuits have features of all three components
(R, L and C), so we have to deal with Impedance,
Z. Phasor diagrams (see textbook for diagrams). *NOTE that when we look
at the RLC AC circuit, that we rotate our previous phasor diagrams,
because "There can be only ONE current." *Minimum impedance
is when purely Resistive or when the two Reactances cancel each other --
the latter is frequency dependent. Can run into problems if expecting
f=60Hz but get f=50Hz or f=25Hz. Phase angle, phi, for resultant V_{max}
vector relative to the I_{max} vector, is phi = tan^{-1}((X_{L}-X_{C})/R).
For impedance matching , where X_{L}=X_{C}, we get the
same equation for the angular frequency omega = 1/SQRT(LC) as for the LC
oscillator -- but what's important this time is that omega is BOTH the LC
oscillator frequency and the AC frequency. In other words, when the
impedance Z is minimized, the LC oscillator part of the circuit isn't
fighting itself. This is why power companies have to worry about
maintaining their frequency -- it affects the impedance of the circuits.
For DC circuits, P = IV. For AC, it is a little more complicated. P_{average}
= I_{rms} V_{rms} cos(phi) -- also P_{average}
= I_{rms}² R -- because of the phase angle between V and I.
(NOTE: Serway's derivation skips a couple of steps, and uses a couple of
trig identies.) For a purely resistive circuit, or one which looks like a
purely resistive circuit, phi = 0°, and so get P_{average} =
I_{rms} V_{rms} .

Quiz 11 Take-Home quiz, handed out on Friday 13 November 2015 and due on Tuesday 17 November 2015, on AC circuit variables. (Click here for a copy. fixed 11-15-15)

Monday 9/7: Labor Day <No Classes>

Tuesday 9/8: Class begins. Intro to Dr Phil. Static electricity. Amber =
elektros in Greek. The Two-Fluid Model of Electricity. Franklin's
One-Fluid Model of Electricity.
Occaam's Razor.
The simple hydrogen atom -- whatever charge is, the charge on the electron
(-e) and the proton (+e) exactly cancel. Electricity & Magentism are
related -- one of the great triumphs of 19th century Physics was the
realization that Electricity and Magnetism were two sides of the same
coin. **Moby Dick** by Herman Melville -- gold coin, lightning and the
reversal of the ship's compass needle. Intro Handout. (Click
here for a copy if you didn't make it to
the first class.)

- Physics Help Room opens today -- 3302 Rood.
- NOTE: PHYS-2080 Lab begins this week. If you missed your first lab, don't panic. You will be responsible, however, to prove to your Lab T.A. that you have read and understood the Introduction to your PHYS-2080 Lab Manual.
**Reminder:**PHYS-2080 is the lab course and is separate from PHYS-2070. I've been asked to remind you to buy the lab manual using the directions emailed to you from Chris Hoffman, the lab supervisor -- he arranged for a special low price for you.

Wednesday 9/9: Real Electric Charges. Two charges: like charges repel, unlike (opposite) charges attract. A Nickel coin has a mass of 5 grams, so about 1/10th of a mole. "Action at a distance" -- Gravity and the Electric Force are not contact forces. The Electric Force between two point charges, Coulomb's Law looks like Newton's Law of Universal Gravity. 1 Coulomb of charge is an enormous amount of charge. Two 1.00 C charges separated by 1.00 meters have a force of nine-billion Newtons acting on each other. Four Fundamental Forces in Nature: Gravity, E & M, Weak Nuclear Force, Strong Nuclear Force. The Hydrogen Atom: Gravity loses to Electric Force by a factor of 200 million dectillion (!!!). The Helium Atom: Putting more than one proton in the nucleus produces enormous forces on the tiny protons -- Need the Neutron and the Strong Nuclear Force (The repulsive force between protons in Helium is about 231N !!!).

FYI: Handout: SI Prefixes and Dr. Phil's Simplified Significant Figures. List of Topics covered in PHYS-2050 when Dr. Phil taught it last.

- Quiz 1 on Thursday for attendance. (Quiz 1A can be downloaded and printed out if you are absent .) Quiz 1½ on Friday, simple problem "to keep us honest".

Thursday 9/10: Distribute Syllabus.
(This is the corrected version.) Conductors (metals) versus non-conductors
(insulators). Conduction electrons in metals -- free to move around.
Insulator electrons very hard to move around. Semi-Conductors sit in the
middle. Sometimes they conduct and sometimes they don't. This means they
act like a switch or valve, and this is the basis for the entire
electronics semi-conductor industry. Charging a conductor by induction. We
need the Strong Nuclear Force to overcome the Coulombic replusion between
protons in the nucleus. Protons repel protons, neutrons don't really stick
to neutrons, but protons do stick to neutrons. Z = # of protons,
determines the element. N = # of neutrons. A = Z + N. **Isotopes:**
Nuclei with same Z, but different N. Hydrogen-1, chemical symbol H (Z=1,
N=0, A=1). Hydrogen -2, Deuterium, chemical symbol D (Z=1, N=1, A=2).
Hydrogen-3, Tritium, chemical symbol T, radioactive (Z=1, N=2, A=3).
Helium-3, chemical symbol He, stable (Z=2, N=1, A=3). Helium-4, chemical
symbol He, extremely stable (Z=2, N=2, A=4). There are about 118 elements
(Z), but tens of thousands of nuclides (Z,N). Quiz 1 was an in-class
Attendance and Check-In form. Q1A coming if you weren't in class.

- No Panic Quiz 1½ on Friday, simple problem "to keep us honest".
- NOTE: Page 14 of the Syllabus has the wrong dates. Need a new Page 14? Click here for a copy..

Friday 9/11: Topic 1 assigned. (Updated Searchable booklist available online NOT UP YET -- use last years for now .) Q1½ In-Class worksheet -- "No Stress Quiz".

Monday 9/14: Finding the net vector electric force F_{E} for a
system of point charges. Remember: In PHYS-2070, Looking at Symmetry and
Zeroes (problems where the answer is zero) as a way of solving problems.
Solving a vector Electric Force problem when there isn't symmetry to
render the problem zero. Example: vector-F_{x} = -3.16 N i-hat,
vector-F_{y} = +4.60 N j-hat. vector-F = 5.581 N @ 124.5°.
Example: q = q_{1} = q_{2} = q_{3} = 2.56 ×
10^{-6} C. d = 2.00 cm = 0.0200 m. vector-F_{2} =
vector-F_{1on2} + vector-F_{3on2} = 0. But vector-F_{3}
does not equal zero. And vector-F_{2} does not equal zero if q_{3}
= -q.

- Review of vectors and vector forces. Review of vector notation for components and Standard Form. Right Triangles and Adding and subtracting vectors: Analytical method. (Check to make sure your calculator is set for Degrees mode. Try cos 45° = sin 45° = 0.7071) Why arctangent is a stupid function on your calculator.

Tuesday 9/15: Find the electric force acting on q_{2} for a
system of three charges. Leaving you to solve for net vector electric
force on q_{1} for problem in
class -- *q _{1} = +1.00 × 10^{-6} C =
+1.00 µC, q_{2} = +2.00 × 10^{-6}
C , q_{3} = +3.00 × 10^{-6} C ,
d = 10.0 cm = 0.100 m*. Give final vector in Standard Form (Magnitude
w/ Units @ Standard Angle), expecting Standard Angle to be just greater
than 90°. (see below)"Action at a distance" -- Gravity and
the Electric Force are not contact forces. The mathematical construct of
the Electric Field. E is not an observable quantity. (Side example:
Methods of measuring speed v, do not directly measure speed v.) Electric
Field is a vector. F

- Solution for Problem given in class for homework on Tuesday.
- Quiz 2 is THURSDAY this week.

Wednesday 9/16: More examples of symmetric 2-D problems for F_{E}
and E-fields. E-field lines radiate *away* from a positive point
charge; converge *towards* a negative point charge. If the universe
is charge neutral, can have all E-field lines from + charges terminating
on - charges.Why use E-fields, when you need the force F = q E anyway?
Because it allows us to examine the environment without needing another
charge. E-field lines allow us to qualitatively sketch what happens when
two charges are near to each other. (1) +q and -q, (2) +q and +q. Very
close to each point charge, the E-field lines are radial outward, evenly
spaced. In the system, the E-field lines interact with each other -- but
E-field lines can never cross. Long range, the system of point charges
looks like a single net charge. The density of E-field lines in an area
gives you an indication of the strength of the E-field line. The numbers
of E-field lines attached to a point charge is proportional to the charge.
E-field lines radiate away from positive charges and terminate on negative
charges. (3) +2q and -q.

- NOTE: Due to doctors' appointments, there are NO 2-4pm Office Hours on 9/16 Wed or 9/17 Thurs.

Thursday 9/17: So far we've looked at Electric Forces and Fields from discrete charges. Now we will look at extended continuous and uniform charges. Direct integration of Electric Force and Electric Field are similar, so we'll just go over direct integration of the E-field. Charge distributions -- lamda (linear charge density, C/m), sigma (surface charge density, C/m²), rho (volume charge density, C/m³). Note the similarity to mass distributions from PHYS-2050. Examples: Rod in-line with line from point P (1-dimensional integration). Q2 in-class.

- NOTE: The class webpage now features some Sample Exam 1s. These are from real PHYS-2070 Exam 1s. Some have solutions, some do not not. The ones without solutions you need to figure out by PTPBIP or comparing answers with others to see if you're right. After all, you don't have the answers in front of you when you take the exams for real. Actual exams may have more white space to work in. Some have more than one exam's worth of material. Obviously there are some problems you can't do yet, because we haven't covered that material yet. Each semester is different, so you wull be told when "the book is closed" on material for your actual Exam 1.
- FORMAT FOR EXAM 1: Problem 1 -- one problem with five parts, much like those in the Quizzes and like the problems in the Sample Exams 1s. Problem 2 -- parts 2a, 2b, 2c and 2e are the Star Problems, which not only require calculus, but to get the Star Points, you must write down the calculus correctly.

Friday 9/18: **Examples**: Rod in-line with line from point P
(1-dimensional integration). Rod
perpendicular to line from point P. Note that in all these cases, we
can predict the long range behavior (E-field behaves as a single point net
charge), and anticipate the close-in short range behavior. Check Serway's
examples (that's your textbook) -- watch out that his notation may be
different. Review of 2-D and 3-D Integration.
Rectangular (area, volume), Polar (circumference, area), Cylindrical
(volume, surface area). Spherical Co-ordinates (volume, surface area,
hollow volume).

- Click on the links above to get the same pages shown in class.

Monday 9/21: Direct integration of Electric Field continued. **Thin
ring of charge** to center point P. (Symmetry!) Thin ring of charge
perpendicular to line from point P. Note that in all these cases, we can
predict the long range behavior (E-field behaves as a single point net
charge), and anticipate the close-in short range behavior. Check Serway's
examples (that's your textbook) -- watch out that his notation may be
different. **Disk of charge** to center point P. Harder to see 1/r²
dependence at long range, but it is clear that E goes to zero. **Electric
Flux**: Electric field times Area. Analogy of a bag or box around a
light, captures all the light rays no matter the size or shape. Note:
Coulomb constant *k* is not the fundamental constant for electricity
-- epsilon-naught is. Coulomb's constant *k*
versus Permitivity of Free Space (epsilon-naught).

- Direct integration to find E for disk on axis shows us how difficult it can be to set up the equations -- there are four kinds of "r" hanging around. We started to set this up in class -- you should finish it Plus you'll need one of the integrals from the back of the book. Page A.30. (May be page A.19 in your textbook.) Table B-5.

Tuesday 9/22: Disk of charge to center point P. Harder to see 1/r²
dependence at long range, but it is clear that E goes to zero. **Electric
Flux**: Electric field times Area. Analogy of a bag or box around a
light, captures all the light rays no matter the size or shape. Use known
E-field of a point charge to evaluate what the Electric Flux must be equal
to. Review of Dot Product.
Gauss' Law for Electricity. Using
Gauss' Law for Point Charge, Conducting
Sphere (case 1: r < R). Note that E-field is zero inside a
spherical conducting sphere (solid or hollow). If the Earth were hollow,
there'd be no gravity inside the Earth either, besides being zero-gee at
center of core. Using Gauss' Law for Point
Charge, Conducting Sphere, Insulating Sphere, Infinite Line of Charge.

Wednesday 9/23: Return Q2. Using Gauss' Law
for Insulating Sphere. The solutions for r>R (outside) and r<R
(inside), must match at r=R. And they do. What happens if the volume
charge density *rho* is not constant? If *rho* only depends on
r, not theta or phi, then when one finds the q_{inside}, one has
to integrate *rho dV*, where rho=rho(r). Q3 in-class.

- NOTE: Direct integration to find E for disk on axis shows us how
difficult it can be to set up the equations -- there are four kinds of "r"
hanging around. We started to set this up in class -- you should finish
it Plus you'll need one of the integrals from the back of the book. Page
A.30.
**(May be page A.19 in your textbook.)**Table B-5.

Thursday 9/24: Gauss' Law for Thick-Walled Insulated Sphere of Charge. A
thick-walled sphere is the volume of the sphere minus the hole in the
middle. Similar for thick-walled cylinders. Gauss'
Law for Infinite Sheet of Charge. P.E. is minus the Work. Potential
V is similar, but the integral is done on E-field not Force. More
importantly the Potential V is an observable quantity. Find components of
E by negative of the partial derivative of Electric Potential function V.
It will turn out that charge accumulates on the tips of long pointy things
-- applies in why some things seem to always get hit by lightning
(golfers, people standing in an open field, church steeples). E_{max}
= 3,000,000 N/C = 3,000,000 V/m, in dry air. Ben Franklin and lightning
rods. Why your hair stands up warning you that you are getting charged.
Handy chart of the four quantities: F_{E} (vector, 2 charges), E
(vector, 1 charge), U_{E} (scalar, 2 charges), V (scalar, 1
charge) .Simplified equation V = E d. (But remember that it's really *delta-V
= - E d .*) Example: Lightning. Equipotential surfaces -- lines of
constant Electric Potential (voltage). *Analogy: Topographic maps, the
equipotential lines are like the altitude contour lines. A skier's line of
maximum descent down a mountain corresponds to the E-field lines. *

**If you are using the Testing Center for Exam 1**, you must (a) make an appointment at the Testing Center AND (b) send me an e-mail saying that you are taking your Exam 1 at the Testing Center at such-and-such a time, so that I know to send an exam over there.

Friday 9/25: Return Q3. Note: It is annoying that electric potential, V,
and its SI unit, V=volts, are both "V". Be careful between units
and variable -- Dr. Phil sometimes spells out the unit as volts. (The
textbooks can get away with this by putting one of the two in italics, but
it is hard to write italics in your notes or on the board.) Simplified
equation V = E d. (But remember that it's really *delta-V = - E d .*)
Example: (1) Can we get sparks from the two slots of an electrical outlet
at 125V? (2) Lightning. Conductor in equilibrium is an equipotential
throughout. In electrostatic equilibrium, E = 0 inside a charged
conductor, but V = constant, not V = 0 automatically (it's *delta V = 0*
inside the conductor). Why charge accumulates on the tips of "pointy
things". Model a conducting blob with a blunt end and a pointy end,
sort of like a piece of candy corn, by a large conducting sphere and a
smaller conducting sphere, connected together by a wire so they are all
equipotentials, i.e. V = constant. For a charged sphere, same as a point
charge: V = kq/r. While the charge on the tip is less than the charge on
the rest, the surface charge density, sigma = q / Area, is much higher. Q4
is a Take-Home on Gauss's Law on a Cylinder, due on Monday
28 September 2015 at the beginning of class (or by 4pm). (Click
here for a copy.)

**Additional Problem:**Click here for a copy.

Monday 9/28: **What is an electron volt (eV)?** (1) It is literally
the charge *e* times 1 volt -- for things with a charge q=±e,
then *qV=(e)(1 volt)=1.602 × 10 ^{-19} J*. (2) Given
the relationship between U

Tuesday 9/29: Last thoughts on the Electric Potential V -- it really is
a scalar, not a vector. Problems with point charges may be zero for E and
not for V and vice versa. Also, can integrate easily over part of the
perimeter of a circle, since it just adds as a scalar. And we can recover
the components of E: *E _{x} = - partial derivative with
respect to x of V*.

Wednesday 9/30: Exam 1.

Thursday 10/1: **Capacitors and Capacitance**. The Capacitor stores
charge +Q on one plate and -Q on second plate, stores energy in the
E-field between the plates. This is different from a battery, which has
energy stored in its chemical reaction. Stories: Dr. Phil & the camera
flash. US Navy seaman vs. the tank capacitor (Cap-2, Seaman-0).
Capacitor Equation. SI unit for
Capacitance is the Farad. 1F is a large capacitor. Usually deal with µF
(microfarad = 1/1,000,000th of a Farad) and pF (picofarad =
1/1,000,000,000,000th of a Farad). Apply Gauss' Law for Electricity to the
constant E-field of the Parallel Plate
Capacitor. We now have an "operational equation", true for
all capacitors, and a "by geometry" equation for the special
case of the parallel plate capacitor. **Example**: A parallel plate
capacitor has plates 1.00 m x 1.00 m and a gap of 10.0 cm. C = 88.5 pF.
*Such a big capacitor with such a tiny capacitance. *Need either
more plate (increase A) or narrow gap (decrease d) to increase C. Work to
assemble charges on a capacitor = Energy stored in the capacitor = U = ½CV²
. Basic circuit diagrams -- elements include wires, battery, capacitor,
and possibly a switch.

- If you missed Exam 1, you need to contact Dr. Phil and arrange to make Exam 1 up.
- NOTE: Just found a typo in my Office Hours. A "2" was missing for 12:50pm -- can't have office hours AND a 1pm class at the same time. Fixed.

Friday 10/2: Two devices connected together in a circuit can only be
connected two ways: series or parallel. In Series,
same charge, share voltage. Equivalent capacitor is always smaller. NOTE:
Remember to take the last reciprocal! In Parallel,
same voltage, share charge. Equivalent capacitor is always larger.
Capacitor Network
Reduction problem. Carefully analyze the network, reducing series or
parallel capacitors to equivalent capacitors, redrawing the circuit each
time. Use table with columns for Q = C V. By going back through the
intermediate diagrams, it is possible to know every value of every
capacitor in the network. Extend the example in class with a fourth
column, U_{E}=½CV², and find the energy stored in the
equivalent capacitor and the sum of the energy stored in all four of the
real capacitors -- if they agree, then our analysis and calculations are
correct -- the battery cannot tell the difference!

- NOTE: Filling in the table is a case where you might want to use 3 + 2 sig. figs., in order to get the ratios to come out really close in the end.
- In Class, we used four 100. pF capacitors.
**Over the weekend, repeat this example**, but with C_{1}= 100.pF, C_{2}= 200.pF, C_{3}= 300.pF, C_{4}= 400.pF.

Monday 10/5: Making a real capacitor. What if not filled with air?
Filling with conductor, must have at least one gap, otherwise will short
outthe plates. A conducting slab inside a parallel plate capacitor makes
two capacitors in series. Charge neutral slab stays charge neutral, but +Q
of top plate attracts -Q on top of slab, and -Q of bottom plate attracts
+Q on bottom of slab. Dielectrics -- an insulator where the +/- charge
pairs are free to rotate, even if they do not move. Dielectric constant
(kappa) and Dielectric strength (E-max). (See Table
26-1, p. 791 ***) Dielectic constant increases capacitance over air
gap. Dielectric strength usually bigger than E_{max} in air. Both
allow you to (a) make bigger capacitors (or smaller for the same values)
and (b) make non-hollow, self-supporting components. Electrolytic
capacitors -- must be connected into the circuit with correct + and -
polarity. Examples of the uses of capacitors and dielectrics. Capacitive
studfinder, uses edge effects of E-field from a capacitor to "see"
the dielectric material behind the wall.

- *** Note that link is not the same dielectric table as the page reference to Serway. I provide it for convenience.

Tuesday 10/6: Examples of the uses of capacitors and dielectrics.
Computer keyboards with switches which have "no moving parts".
Capacitative stylus for tablets. Biometric security devices. **Electrostatics
(equilibrium) to Electrodynamics (moving charges)**. Current defined: i
= delta-Q/delta-t = dq/dt. (1) The net charge of a current carrying wire
remains zero, so have 1.00 A = 1.00 C/sec isn't the same as having 1.00 C
of bare charge lying around. (2) Positive charges moving in same direction
as a positive current is the same as negative charges moving the other
way. (3) When people say electricity moves "at/near the speed of
light", it does not mean the electrons in the wire are moving at the
speed of light. It is the E-field which is moving at the speed of light in
the material. (4) Discussion of microscopic theory of charges in a
conductor. Drift velocity is the very slow net movement of the electrons
moving randomly in the wire. See pp. 753-755. Drift velocity of electrons
in copper wire is about 2.23×10^{-4} m/s. This microscopic
theory becomes more important as we go to smaller and smaller circuit
elements in our microchips.
Moore's Law. (5)
The charges are moving in response to an E-Field, because we have a
non-zero delta-V. We can have an E-field and delta-V inside a conductor in
a circuit because this is no longer an electrostatic equilibrium problem.

Wednesday 10/7: Q5 in-class.

Thursday 10/8: **The Simplest Circuit:** Battery, wires, load
(resistor). **Resistance vs. Conductance**.
Ohm's Law: V=IR form. (Ohm's "3
Laws") We usually treat the wires in a circuit as having R=0, but
they usually are not superconductors. Resistance is a function of
temperature. Kammerleigh Onnes 1916 work on extending the R vs. T curve
toward T = 0 Kelvin. Discovered Superconductivity, where R=0 identically.
**Joule Heating, Power Law:** P = IV (also 3 forms). Resistance by
geometry. R = rho (L / A), where rho = resistivity of the material, L =
length and A = cross-sectional area. Continuing with Simple Circuits...
Series and Parallel Resistors: Two
devices connected together in a circuit can only be connected two ways:
series or parallel. In Series, same current, share voltage. Equivalent
resistance is always larger. In Parallel, same voltage, share current.
Equivalent resistance is always smaller. Resistor Network Reduction.

- NOTE: The Series and Parallel rules for resistors are the exact opposite of the capacitor rules.
- First Sample Exam 2s posted on class web page.

Friday 10/9: Return X1. Continuing with Simple Circuits...
Series and Parallel Resistors: Two
devices connected together in a circuit can only be connected two ways:
series or parallel. In Series, same current, share voltage. Equivalent
resistance is always larger. In Parallel, same voltage, share current.
Equivalent resistance is always smaller. **Resistor Network Reduction**.
(Similar rules to Capacitor Network Reduction except "opposite".)
For example given in class, Resistor R1 sees the largest current and
dissipates the largest amount of energy per second (Power in Watts). This
means it is also the most vulnerable. (Story of radio "repair"
call from 4,000,000,000 miles.)

- NOTE: Filling in the table is a case where you might want to use 3 + 2 sig. figs., in order to get the ratios to come out really close in the end. (Same instruction as with capacitor reductions.)
- In Class, we used four 100. ohm resistors.
**Over the weekend, repeat this example**, but with R_{1}= 100.ohm, R_{2}= 200.ohm, R_{3}= 300.ohm, R_{4}= 400.ohm. Indentify the most vulnerable resistor.

Monday 10/12: **Ohmic versus non-ohmic materials:** If R=constant
over operating range, then we say the material is "ohmic". If R
is not constant, it is "non-ohmic". **Example**: Because of
the temperature dependence of R, the filament of an incandescent light
bulb has a very different R when lit or dark. Therefore measuring the
resistance of a light bulb with an ohmeter is useless. The ohmeter
measures resistance using a small reference voltage. The ohmeter does not
get hooked up to the live circuit -- if you want to find resistance *in
situ*, measure V and I, then use Ohm's Law. This always works. Discuss
incandescent (glowing filament) light bulbs and modern **Christmas tree
lights**. Bulbs in series, but each bulb is in parallel with a small
resistor, to keep whole string lit when one bulb burns out. Older sets
were either parallel (full 120 volts) or series (one burned out bulb takes
out the whole set). Newest sets use LED lights, much less current, much
longer estimated lifetimes. **Real batteries** consist of a "perfect"
battery (Electromotive force = emf, or script E) in series with a small
**internal resistance**, r. As chemical reaction in battery runs down,
the internal resistance increases. V = (emf - i r) = i R. Don't cut open
batteries. Comments on different types of disposable (carbon-zinc,
alkaline, lithium) and rechargable (Rayovac Renewal alkaline, NiCad, NiMH,
Li-ion) batteries. Multi-cell batteries (6V lattern battery, 9V
transistor/smoke alarm, 510V dry cell). **Tip for weak car battery on
cold day**: Run headlights for 30 to 90 seconds. High internal
resistance will warm the battery and make it more efficient. Not all
circuits can be reduced by serial and parallel network analysis.

**Homework from over the weekend:**In Class, we used four 100. ohm resistors.**Over the weekend, repeat this example**, but with R_{1}= 100.ohm, R_{2}= 200.ohm, R_{3}= 300.ohm, R_{4}= 400.ohm. Indentify the most vulnerable resistor. R_{1}must still see the highest current, since it is the only resistor directly connected to the battery. But now R_{3}sees the highest power due to Joule heating. That makes R_{3}the most vulnerable resistor. If I was overengineering this circuit, I would use better resistors for R_{1}and R_{3}, or create a quad parallel/series equivalent resistor replacements for R_{1}and R_{3}.

Tuesday 10/13: **How to jump a car battery** correctly and safely.
(Improper jump can result in hydrogen explosions, boiling sulfuric acid,
etc.) **Kirchhoff's Laws: **(1) The sum of all currents in and out of
any junction must be zero. (2) The sum of all voltage gains and voltage
drops about any closed loop is zero. Practically speaking, if there are N
junctions, then (1) will give you (N-1) unique equations, and if there are
M loops that can be made in the circuit by going around the perimeter of
each "puzzle piece", then (2) will give you sufficient unique
equations. You will get the same number of equations as you unknown
currents through the resistors. NOTE: EE students and those who have had
ECT-2100 (?) may know a "better" way to solve Kirchhoff's
problems. But the brute force algebra approach has the advantage of being
based on the Physics, so has instructional value.
Example in class
with two batteries and 3 resistors had 3 equations in 3 unknowns. Solution
by brute force algebra here.
Note that our assumption that i_{2} goes to the left was wrong,
because the solution gives i_{2} as slightly negative. Minus
signs mean "other direction". In this case, there is a small
current going backwards through the smaller battery, V_{2}.
Perhaps this is a charging circuit for a rechargeable battery?

- To jump a car, you have a good battery and a bad battery. Both have a
positive terminal and a negative terminal. Jumper cables consist of two
heavy duty cables held together -- the ends have big spring loaded clamp
jaws. One wire had red (positive) clamps, the other black (negative)
clamps. Make sure that bumpers of cars don't touch -- and that you
aren't touching anything metal on both cars at once.
**Order of connection:**(1) Positive wire to positive terminal of dead battery. (2) Positive wire to positive terminal of good battery. (3) Negative wire to negative terminal of good battery. (4) Last connection DOES NOT go to negative terminal of bad battery, but to something metal under the hood that does not look like it would move when the engine starts AND is some distance from the bad battery. On GM cars, I look for a big bolt with a big wire on the end of the generator -- that's the ground. The hood latch is dirty and greasy and won't work. - Or call AAA or other road assistance and have them deal with it.
- Note: the solution to the Kirchhoff Law problem had resistors of 10.0 ohms, 20.0 ohms and 30.0 ohms -- not 100 ohms, 200 ohms, 300 ohms -- but the algebra is the same.

Wednesday 10/14: Q6 in-class.

Thursday 10/15: Example
in class had 3 equations in 3 unknowns -- (1) i_{3} is
opposite direction than in class, (2) resistors are divided by 10 from
class.. (Solution by brute force algebra
here). Example: A case with
five resistors, where some arrangements can be reduced by series/parallel
reduction, others only by Kirchhoff's laws. The special case with symmetry
and R_{1}=R_{2} and R_{3}=R_{4}. Then
the voltage on either side of R_{5} is the same and the voltage
different is zero -- that means no current through R_{5} and it's
as if R_{5} doesn't exist. With the troublesome resistor gone,
you can reduce by series/parallel reduction. Note that getting "rid"
of a resistor means using an eraser -- replace with a plain wire
introduces a new circuit element and is NOT the same. **RC series
circuit.** Use Kirchhoff's 2nd Law to get a loop equation for voltage
gains and drops around charging capacitor. q=q(t) and i=i(t)=dq/dt means
that we can use calculus to find the current through the resistor and the
charge on the capacitor. Calculus derivation of q(t) for
charging capacitor and discharing circuits.

Friday 10/16: **RC series circuit.** Use Kirchhoff's 2nd Law to get
a loop equation for voltage gains and drops around charging capacitor.
q=q(t) and i=i(t)=dq/dt means that we can use calculus to find the current
through the resistor and the charge on the capacitor. Calculus derivation
of q(t) for charging capacitor and
discharing circuits. RC current i(t) will be the same in both cases. Who
knew that (ohms) × (farads) = (seconds)? By time t=3RC, a charging
capacitor will reach 95% of its top charge, or a discharging capacitor
will be down to 5% of its original charge. Either way the current will be
down to 5% of its maximum value. Solve q(t) charging or discharging
equation for *t*, using the natural log (ln) function to get rid of
the exponential. Applications for slowing down the charging/discharging of
a capacitor. Backup power (short duration) -- not quite the same as a
battery. Next week: Building an ammeter and a voltmeter -- does the very
act of taking a measurement, change the thing you are trying to measure?
Q7 Take-Home on Kirchhoff's Laws, due Monday 26 October 2015. (Click
here for a copy)

- Useful to have e
^{-0}, e^{-1}, e^{-2}and e^{-3}and (1-e^{-0}), etc. for 1RC, 2RC and 3RC.

Monday 10/19: **Measurement: **Building an ammeter or voltmeter --
non-digital version with a needle. The Galvanometer is a generic meter. It
has a resistance and the needle moves in response to a current through a
tiny coil. Since meters must be connected to the circuit, technically they
change the circuit. However, we will show that the design of an ammeter
and a voltmeter minimizes these changes. Ammeters measure current by
connecting in series to the circuit. Voltmeters measure potential
difference by connecting in parallel to the circuit. The Galvanometer is a
generic meter. It has a resistance *R _{G}* and the needle
moves in response to a current through a tiny coil. The full-scale
deflection current,

- NOTE: The numbers we found for the 5.00 A
ammeter and 5.00 volt voltmeter resistors were: r
_{s}= 0.001262 ohms and R_{v}= 49,940 ohms. The galvanometer had a resistance R_{G}= 63.1 ohms and a full-scale deflection current i_{FS}= 1.00 ×10^{-4}A. The high resistance wire we used for the shunt resistor in the ammeter had an R/L = 0.000147 ohm/cm. For I = 5.00 A and V = 5.00 volts, the load resistor would be R = 1.00 ohms. - Homework: Consider our Simplest circuit with a 5.00 volt battery and a 1.00 ohm resistor, with BOTH our ammeter and voltmeter attached. Find the equivalent resistance and see how the voltage and current for the load resistor changes. In other words, does the act of metering affect the meter reading?

Tuesday 10/20: Return Q5, Q6. HW problem still pending... can't talk
about effect of measurement on a circuit... J-vector = sigma ×
E-vector (current density = conductivity × E-field) is the vector
version of Ohm's Law, where J-vector is the Current Density (although we
assign a direction, technically current is a scalar and current density (I
/ Area) is the vector quantity). There are at least two Sample Exam 2
problems with J-vector. "**Magnetism is just like Electricity, only
different.**" Most people are familiar with (1) magnets sticking to
some metals, not others such as stainless steel and (2) if you have two
magnets, they may attract or repel. North and south are analogous to plus
and minus charges. Real Magnets are dipoles (North and South ends,
linked). Break a magnet in half, and you either get two new magnets -- or
nothing. So far, there is no evidence that there are Magnetic Monopoles
(magnetic charges: *q _{M}* , isolated North or South
poles). B-field = magnetic field. B-field of a bar magnet resembles the
E-field of an electric dipole (+q and -q separated by a distance).

- Using V = Ed and R = rho (L/A), we can turn J-vector = sigma × E-vector into V = IR. It's all Ohm's Law.
- "The Data Point" -- the one possible candidate for a magnetic monopole. Or not.

Wednesday 10/21: "**Magnetism is just like Electricity, only
different.**" Most people are familiar with (1) magnets sticking to
some metals, not others such as stainless steel and (2) if you have two
magnets, they may attract or repel. North and south are analogous to plus
and minus charges. Real Magnets are dipoles (North and South ends,
linked). Break a magnet in half, and you either get two new magnets -- or
nothing. So far, there is no evidence that there are Magnetic Monopoles
(magnetic charges: *q _{M}* , isolated North or South
poles). Rules similar to Electric Charges: Unlike poles attract, like
poles repel.

**If you are using the Testing Center for Exam 2**, you must (a) make an appointment at the Testing Center AND (b) send me an e-mail saying that you are taking your Exam 2 at the Testing Center at such-and-such a time, so that I know to send an exam over there.

Thursday 10/22: Magnetic Force on a Moving Electric Charge - The Cross Product and Right-Hand Rule (R.H.R.). The Cross Product (or Vector Product) is the exact opposite of the Dot Product (or Scalar Product). Multiplying two vectors together by a cross product gives us another vector (instead of a scalar). And the cross product is not commutative, vector-A × vector-B = - (vector-B × vector-A), so the order is paramount. Using Right Hand Rule to assign directions to x,y,z coordinates. Constant speed, perpendicular constant magnetic force --> Uniform Circular Motion. Cyclotron frequency -- no dependence on the radius (constant angular velocity). Velocity Selector - the Magnetic Force is speed dependent, the Electric Force is not. So we can use an E-field to create an Electric Force to cancel the Magnetic Force on a moving charged particle, such that at the speed v = E / B, the particle travels exactly straight with no net force -- any other speed and the particle is deflected into a barrier. Hence a velocity selector "selects" velocities...

Friday 10/23: Velocity Selector - the
Magnetic Force is speed dependent, the Electric Force is not. So we can
use an E-field to create an Electric Force to cancel the Magnetic Force on
a moving charged particle, such that at the speed v = E / B, the particle
travels exactly straight with no net force -- any other speed and the
particle is deflected into a barrier. Hence a velocity selector "selects"
velocities... Velocity Selector. Mass
Spectrometer - different semi-circular paths for ions of different
mass but same velocity. Can determine chemicals, molecules, and separate
isotopes (same element, different number of neutrons in nucleus, so
different mass -- cannot be separated by ordinary chemical means). Mass
Spectrometer as Calutron -- detecting or separating isotopes, something
that cannot be done by ordinary chemical means. *NOTE:
The book is effectively closed for Exam 2 topics now.* UPCOMING
TOPICS: Since there is a magnetic force on a moving
electric charge in a magnetic field, and a current carrying wire is a
collection of moving electric charges, there must be a magnetic force on a
current carrying wire in a perpendicular B-field. Later we will see that a
current carrying wire creates a B-field, so there ends up being a magnetic
force between two current carrying wires. **A current carrying
wire** consists of moving electric charges, and so therefore would see a
magnetic force from a magnetic field. Discussion of microscopic theory of
charges in a conductor. Drift velocity is the very slow net movement of
the electrons moving randomly in the wire. Magnetic
Force on a Current Carrying Wire. Technically current is not a
vector, despite the fact we talk of direction of vector-J = current
density = current/cross-sectional area -- the vector related to current.
Since we want to use the current I, we use the displacement vector L for
the direction.

Monday 10/26: Magnetic Force on a Current
Carrying Wire. Demo -- hey it works and even in the right direction!
Technically current is not a vector, despite the fact we talk of direction
of current. J = current density = current/cross-sectional area is the
vector related to current. *NOTE: J-vector = sigma × E-vector
(current density = conductivity × E-field) is the vector version of
Ohm's Law. This was NOT on Exam 2*. So we use the displacement vector L
for the direction. For a Closed Loop, the net Magnetic Force from a
constant B-field is zero. Magnetic Torque on
a Current Carrying Wire. We use the enclosed area vector A, whose
direction is defined by using the Mode 2 R.H.R. (fingers curled around the
direction of the current loop, thumb is the area vector A perpendicular to
the plane of the loop). Left as is, this system is an oscillator -- the
torque goes to zero after 90° and then points the other way. But if
we can reverse the direction of the current after the torque goes to zero,
then the rotation can continue -- and we have a primitive DC electric
motor. Q8 Take-Home due.

Tuesday 10/27: **Hall Effect** -- a device with no moving electrical
parts -- proves that charge carriers in a current carrying wire are
negative, not positive. "The 200 Year Hall Effect Keyboards",
will last "forever", but made obsolete in two years when Windows
95 added three keys. Gauss' Law for
Magnetism. Not as useful as Gauss' Law for Electricity, because it
is always zero (no magnetic monopoles). It turns out that a moving
electric charge or a current carrying wire creates a magnetic field. This
makes sense that a moving charge or current would have a magnetic force
from an external magnetic field, if its own magnetic field is interacting
with the external field, just as two charges "see" each other
via their electric fields. **The permeability of free space**, µ_{0},
is the fundamental constant of magnetism. It is unusual in that we know
the exact mathematical representation, which is why is given as 4pi ×
10^{-7} T·m/A. If we calculate 1/sqrt(epsilon_{0} ×
µ_{0}), we get the *c* = speed of light in vacuum --
again showing the fundamental connection between electricity and
magnetism. Review for X2.

Wednesday 10/28: Exam 2.

Thursday 10/29: The Biot-Savart Law.
B-field from a infinitely long straight
current carrying wire by direct integration. (Serway has a similar
example, but rather than do the integral in *x*, he does this theta
substitution which Dr. Phil does not think is straight forward.) Circular
loop of current carrying wire by integration for *P* at the center
of the loop. (Serway's example allows for *P* to be on a line
perpendicular to the loop.) B-field for a circular current carrying wire
at the center -- or any part of a circle. Note that a wire coming in along
the r-hat direction makes no contribution to the B-field.
Magnetic Field loops from a Current Carrying
Wire. RHR has "two modes". Mode 1 uses three mutually
perpendicular directions for when you have three vectors (A × B = C
is 1-2-3, x-y-z). Mode 2 uses the curling of the fingers to represent the
circulation of a field or the motion of a current, etc., with the thumb
representing the relevent vector or direction. Magnetic
Force between Two Current Carrying Wires. Combining problems, we
find that for two **parallel current carrying wires**, with the
currents in the same direction, the magnetic field from wire 1 creates an
attractive magnetic force on wire 2. And the magnetic field from wire 2
creates an attractive magnetic force on wire 1. (Two forces, equal and
opposite, acting on each other -- this is exactly as it should be with
Newton's 3rd Law.) **Anti-parallel currents **(wires parallel, but
currents in opposite directions) repel. **Crossed currents** (wires
perpendicular to each other) see no magnetic force on each other.

- NOTE: No Thursday 2-4pm Office Hours due to doctor's visit.

Friday 10/30: Magnetic Force on a Current
Carrying Wire. **Demo** -- hey it works and even in the right
direction! **Operational defnition of the ampere and the coulomb**:*
If the Force per length for two wires with a current I separated by 1
meter is F/L = 2 × 10 ^{-7} N/m, then I = 1 A exactly. Then
in 1 second, 1 C of charge is moved by this 1 A current. *Gauss'
Law for Magnetism. Not as useful as Gauss' Law for Electricity,
because it is always zero (no magnetic monopoles). However, there is
something we can use in a similar way which involves involving a path
integral along a B-field and the current(s) contained inside --
Ampere's Law. Use in a way similar to the
way we used Gauss' Law for Electricity. Use symmetry and geometry to
select your Amperean Loop to your advantage. 3-D directions and R.H.R.

- Your Weekend Moment of Pure Physics Zen: Article about the Video.
- Time Change on Sunday! 2am Eastern Dayling Time magically becomes 1am Eastern Standard Time. Adjust your clocks accordingly.

Monday 11/2: Ampere's Law. Use in a way
similar to the way we used Gauss' Law for Electricity. Use symmetry and
geometry to select your Amperean Loop to your advantage. **B-field of a
Solenoid**. (*NOTE: The integrals for the L and R sides of the
Amperean Loop for Ampere's Law are zero because: (1) the B-field is zero
outside the solenoid and (2) for that part of the path which is inside the
solenoid, the B-field and the ds-vector are perpendicular, so the dot
product is zero as well.*) **Edge effects:** E-field of parallel
plate capacitor vs. B-field of solenoid. More Comments about making a real
velocity selector -- trying to stuff a capacitor for the E-field and a
solenoid for the B-field in the same space! Comments about making real
coils. Insulating varnish, heat damage. Yields affect time and money. **B-field
of an infinite sheet of current**. The B-fields are constant and
parallel to the surface and perpendicular to the current elements.
(Similar to the E-field of an infinite sheet of charge.) Note that
Ampere's Law has a flaw, which we will correct at a later date. If a
current carrying wire can create a magnetic field, can a magnetic field
passing through a coil create an electrical current? Demo: Magnet moving
into a coil, causing current to flow through galvanometer.
Faraday's Law of Induction. A changing
magnetic flux induces a current, induces an e.m.f., in the circuit,
substituting for the battery as the power source. Lenz's Law "of
maintaining the status quo." The coil acts as if it opposes any
change of the magnetic flux inside, by inducing a magnetic field to cancel
and increasing flux or maintain a decreasing flux. To create this induced
magnetic field, one needs an induced current, which is powered by an
induced *emf*. Turn a coil in a magnetic field and the flux changes,
thereby inducing a B-field, emf and current. Has same 180° problem
that a DC motor has.

- First Sample Exam 3s posted. Remember, Exam 3 is on TUESDAY 24 November 2015. No class the next day...

Tuesday 11/3: **Demos**: Cow magnets -- powerful cylindrical,
rounded end magnets which get dropped into a cow's first stomach, to
collect nails, bits of barbed wire, etc. from continuing on to the cow's
other stomachs. **Demo**: Magnet moving into a coil, causing current
to flow through galvanometer. **Demo**: Lenz Law race between cow
magnets dropped through (a) plastic pipe, (b) non-magnetic aluminum pipe
and (c) non-magnetic copper pipe. Something is going on, such that the
magnets travel much slower through the metal pipes -- and the thicker
copper pipe was much slower than the thinner aluminum pipe. Induced
B-fields due to changing B-fields of falling magnets are created by
induced currents and induced emf -- as the magnet enters and leaves a
circular region of metal pipe, it is slowed by magnetic forces between its
magnetic field and the induced B-field. Ford test electric vehicle with
**inductive charger** -- no exposed metal contacts, everything covered
in smooth plastic. Dr. Phil has been advocating for 69 semesters that
small charging bricks for all our electronic devices be replaced with
universal charging-by-induction systems. Beginning to see first practical
systems such as Powermat and some new non-Apple cellphones. **(2)
Safety: **(First, how regular fuses and circuit breakers work -- and why
that isn't fast enough to prevent some types of accidents.)

- Alas, the Jumping Rings demo didn't work. However, I've put in a repair order, so hopefully we'll see it soon.
- Powermat Wireless Charging System.
- DON'T TRY THIS AT HOME: I have never heard of Coin Shrinking before, but it seems to me that they are getting extremely large induced currents flowing in these coins, something in the 750,000 A range, which probably heats them to be soft enough that the attraction between parallel current paths might cause the coins to contract. Quarters shrunk down to the size of dimes? I suspect that the reason it didn't work so well with a nickel, is that nickel is a pretty hard metal and has a higher melting point. So they probably didn't get a big enough current, so instead of shrinking the nickel, they just sort of wrinkled it. (grin).

Wednesday 11/4: **(2) Safety: **(First, how regular fuses and
circuit breakers work -- and why that isn't fast enough to prevent some
types of accidents.) **Ground Fault Interupt** -- if the current
doesn't return via the return wire, because it has found another
conductive path, then the 2 wires (hot and return) total a
net-enclosed-current for Ampere's Law, generating a B-field in a metal
ring, detected by an induction coil wrapped around the ring and this sets
off the relay which breaks the circuit. Q9 in-class.

Thursday 11/5: Return X2. Practical uses for induction: **(1) Heating:
The Good** -- Heating the bottom of a metal cooking pot by induction:
New type of cooking range uses sealed induction heating elements instead
of exposed hot resistors or open gas fed flames -- usable for metal pans
only. **The Bad **-- A slab of metal used to conduct a B-field can
waste energy as heat if the B-field is changing, such as in an AC circuit.
By making the slab out of thin plates insulated from each other, the
B-field still can go around the metal, but the perpendicular loops of Eddy
Currents can only have a diameter equal to the thickness of the metal.
Small eddy currents cannot generate much heat because induced emf is too
small. **Why an Inductor has Self-Inductance **-- running a current
through a coil creates a magnetic field and therefore changes the magnetic
flux in the coil. The inductor has to respond to that change. Inductance
can be a big deal. Even our Simplest Circuit (a resistor hooked up to a
battery) forms a loop, and the loop must respond to the circuit being
turned on.

- FYI: Remember, last day to withdraw from a class is Monday 9 November 2015. BUT... should you decide to, make sure you CAN withdraw before you do so -- sometimes it messes up scholarships or student visas if you drop below a certain number of credits.
- Full estimated course grades will be posted tomorrow. (There is still plenty of time to pull your grade up -- see Dr. Phil)

Friday 11/6: **Why an Inductor has Self-Inductance **-- running a
current through a coil creates a magnetic field and therefore changes the
magnetic flux in the coil. The inductor has to respond to that change.
Inductance can be a big deal. Even our Simplest Circuit (a resistor hooked
up to a battery) forms a loop, and the loop must respond to the circuit
being turned on. Practically speaking, you cannot have a purely inductive
circuit with just a battery and L, you really have some resistance as
well. L = self-inductance, SI units
(Henry) = (H). Use Self-Inductance, Faraday's Law and the solenoid
equation to generate the by-geometry
equation for an air-filled solenoid. **Series
RL Circuit**, similar to Series RC Circuit, except that energy is
stored in the magnetic field at the maximum current. U_{L }= ½
L I ². RL Circuit for energizing the
coil. Equation for current *i(t)* for energizing the coil, is
similar in appearance to the equation for *q(t)* for a charging
capacitor. Now we will de-energize the coil. *NOTE: In class I used my
usual brute force approach, instead of Serway's. The Kirchhoff Loop for
the de-energining coil is -iR -L(di/dt) = 0. Since the current is
decreasing, di/dt is negative*** and the induced emf from the coil becomes
a voltage gain*. Solution for the de-energizing coil, *i(t)* is
the same form as the current *i(t)* for discharging capacitor.
Solution for the magnitude of the induced emf, script-E_{L}
= -L di/dt, is the same for both energizing and de-energizing circuit,
much like the current i(t) = dq/qt gives the same solution for both
charging and discharging RC circuit. RL Circuit, similar to RC Circuit,
except that energy is stored in the magnetic field at the maximum current.
U_{L }= ½ L I ². **LC Oscillator circuit.** Same
2nd order differential equation as the Simple Harmonic Oscillator
(PHYS-2050), such as a mass on a spring. Solutions
are sines and cosines. Energy is held constant for all *t*
between the capacitor and the inductor. Can't really have a true LC
oscillator, since normal wires and coils have a resistance which
dissipates energy through Joule heating. LC Oscillator solution: q(t) = Q_{0}
cos(omega t + phi), where *omega = 1 / SQRT (LC)* is the angular
frequency and *phi* is a phase angle. Energy is held constant for
all *t* between the capacitor and the inductor. U = U_{C }+
U_{L }= q²/2C + ½Li² = Q²/2C = ½ L I ².

- *** Note: The mistake in class was that for the Series RL circuit, I just needed to note that di/dt is negative -- I did not need to "fix" the minus sign. This is different from the LC oscillator circuit, where i = -dq/dt for the discharging capacitor and I needed to add in the additional minus sign. Clear? (grin)
- New Mid-Term Grades posted.
- NOTE: Series RL current has been fixed. (11-11-2015 We)