6. (Optional) The circuit shown is linear and time-invariant. a. Write the differential equation with vc as the dependent variable, and indicate theproper initial conditions as functions of vc(0), and i¿(0). Hint: Write a KCL atnode 1, expressing i, in terms of vc and its derivative, and then a KVL in termsvc as an independent variable. b. Calculate the zero-input response vc(t), and i¿(t). Assume i¿ (0) = 1A, and%3D v_{c}(0)=4 V . R_{1}=4 \Omega, R_{1}=2 \Omega, L=1 H, \text { and } C=\frac{1}{2} F

Insulator Conductor Semiconductor Intrinsic semiconductor Extrinsic semiconductor N-type semiconductor P-type semiconductor P-N Junction Forward biased Reverse biased Drift Current Diffusion Current Bipolar Junction Transistor (BJT) NPN PNP metal-oxide-semiconductor field-effect transistor (MOSFET) NMOS PMOS Diode-transistor (DTL) logic Resistor-transistor (RTL) logic Transistor-transistor (TTL) logic Emitter-coupled (ECL) logic Complementary MOS (CMOS) logic Bipolar CMOS (BICMOS) logic

3.51 For the vector field D = R3R², evaluate both sides of the divergence theorem for the region enclosed between the spherical shells defined by R =1 and R = 2.

Hot water at an average temperature of 70°C is flowing through a 15-m section of a cast iron pipe (k- 52 W/m K) whose inner and outer diameters are 4 cm and 4.6 cm, respectively, The outer surface of the pipe is exposed to the cold air at 10°C in the basement, with a heat transfer coefficient of 15 W/m2-K. The heat transfer coefficient at the inner surface of the pipe is 120 W/m2-K. Ignoring radiation determine the rate of heat loss from the hot water in W.

A dipole and a solid sphere of charge +Q are oriented as shown in Figure. The dipole consists of two charges q and - q, held apart by a rod of length s. The center of the dipole and the sphere are at a distance d from the location A. q = 6 nC, s =4 mm, d = 14 cm, and Q = 8 nC. \text { Find the magnitude of the electric field }\left|\overrightarrow{\mathbf{E}}_{\text {dipole }}\right| \text { due to the dipole at the location } \mathbf{A} . (10 points) Draw the direction of the electric field due to the dipole at the location A and write the electric field as a vector Edipole· \text { 5) Find the net electric field vector } \overrightarrow{\mathbf{E}}_{\text {net }} \text { at location } \mathbf{A} \text { due to dipole and the solid sphere. } \text { ) If a proton is placed at location } A, \text { what would be the net electric force vector } \vec{F}_{\text {net }} \text { on the proton? } \text { Find the electric field vector } \overrightarrow{\mathbf{E}}_{\text {Solid sphere }} \text { due to the solid sphere at the location } \mathbf{A} \text {. }

A periscope consists of two 90-45-45º prisms (as shown in Figure) are made of lossless glass with n=1.5. The reflection off the inclined planes are 100% (total reflection per Snell's Law), i.e. |T|= 1. (a) What power(in dB) is the transmitted signal through the bottom prism relative to the input? (b) If the incoming signal has a magnetic field given by: H,(x,f) = 23.77 sin(10"r-kx)mA/m, what is the first reflected E-field? What is k?feld B

1) Triboelectric Series: Find a couple of pairs of materials from the triboelectric series and rub them together. Do they attract each other as advertised? Does the strength of attraction depend on how hard or for how long they are rubbed together? Does the attraction wane with time? Why? Which material is is positive and which one is negative? What practical uses can you imagine for this phenomenon? What dangers? 2) Drying Clothes: Dry some clothes in a tumble dryer or sneak into a laundromat and examine someone else's tumble dried clothes. Find some items that stick together and check their labels for their compositions. What do you see as you pull the clothes apart? What do loose threads do? Why? How is this behavior explained by their composition? What do you hear as you pull them apart? Why? Do wet clothes stick together this way or not? If so,why and if not, why not? This so-called static cling is really tribo electric electrostatic cling.Why do you suppose advertisers call it just static cling? How do anti-static dryer balls and dryer sheets work? Any additional observations you can make are welcome. 3) Scotch Tape: Pull out a couple of stretches of scotch tape from a roll. Is the tape attracted to your skin? How about to glass or metal? Why? Do the two stretches of tape attract or repel each other? Why? Other observations are welcome. 4) One of Your Own: Find your own example of triboelectricity in your home. Experiment with it and describe what you find. Quantitative observations and measurements are welcome welcome.5) Shuffling Shoes: Shuffle your shoes across a carpet and then touch a door knob or some electrically grounded object such that you get a small electrical shock. You might see a small electrical arc. (Better yet, touch the back of someone's ear who's not expecting it,but be prepared to get yelled at.) Explain the shock in terms of the tribolectric series. Did your shoes charge positive or negative? Modern carpets have special chemicals to reduce this effect? How might these antistatic agents work? It's been proposed that this effect can be eliminated by weaving metallic threads through the carpet. Why? 6) Swiffer: How do Swiffer dusters work from the point of view of triboelectricity and induction? 7) Dusty Fan Blades: Find an old fan and look at its blades. Very likely they will have a layer of dust on them. Why should dust settle there, considering that the fan blades turn very fast and, therefore, should shake the dust loose? What holds the dust on the blade sand how did it settle there in the first place? 8) Sparkin' Obtain some wintergreen lifesavers that have some real sugar in them – avoid artificial sweeteners. Go into a dark room, let your eyes adjust for about a minute, then crush the lifesaver quickly. A pair of pliers is ideal, but crunching them in your mouth will also work. If you use your teeth, perhaps do it in a bathroom facing a mirror so you can see into your mouth as the lifesaver is quickly crushed. What do you see? What do you think is going on? In your grandparents' generation, this was a popular pass time: going up to lover'slane and sparkin'.

5.7 (note: TRL = 0 because it is not given in the problem statement)

3.46The scalar function V is given by \text { (a) Determine } \nabla V \text { in Cartesian coordinates. } V=\frac{2 z}{x^{2}+y^{2}} (b) Convert the result of part (a) from Cartesian to cylindrical coordinates. (c) Convert the expression for V into cylindrical coordinates and then determine VV in those coordinates. Compare the results of parts (b) and (c).

In the shown figure, C, = C, = Co- 18.6 nF and C, = C = C, = 10.8nF. The applied potential is Vab= 54 v I. What is the equivalent capacitance of the network between points a and b? II. Calculate the charge on the equivalent capacitor ?