PHYS 122 SI Tyler Griffith Claire Daugherty 1:00pm-2:30pm 12/11/2022 tsg35@case.edu ctd35@case.edu :) ! Final Review ! (: Concepts: Electrostatics Coulomb’s Law Insulators and Conductors Charge Distributions Gauss’ Law Electric Potential Capacitance Dielectrics Electric Energy Advice: Draw Diagrams! Magnetism Ampere’s Law Biot-Savart Law Generalized Lorentz Force Right Hand Rules Faraday’s Law Inductance Solenoids Magnetic Energy Circuit Applications Components Batteries Resistors Capacitors Inductors Series & Parallel Properties Kirchhoff's Rules (with faraday’s) LR & RC Start From Most General Principles! Concepts covered in this review: 1. LR and RC circuit analysis 2. Long term/short term behavior of inductors and capacitors 3. Faraday’s Law and Lenz Law 4. Conductors 5. Gauss’ Law 6. Superposition 7. Potential 8. Kirchhoff’s Laws 9. Multiloop Resistor Circuit Analysis Look for Symmetries! 1. Circuit Analysis and Faraday's Law a. In circuit 1, find the current through the circled resistor immediately after the switch is closed. b. In circuit 1, find the current through the circled resistor a long time after the switch is closed. c. In circuit 2, find (i) the current Ii through the resistor immediately after the switch is closed, (ii) the current If through the resistor a long time after the switch is closed, (iii) The characteristic time constant T to go from situation (i) to situation (ii). d. Now consider another circuit (circuit 3) that happens to be nearby circuit 2 as shown (perhaps they are different circuits on one computer chip). The actual components in circuit 3 (not shown) are not important, but taken together they have an effective resistance Req. Circuit 3 is in the shape of a single square loop, with side length l. Find the mutual inductance M between circuit 2 and circuit 3. You can assume that the magnetic field experienced by circuit 3 only comes from the nearest wire of circuit 2, and that that wire can be considered as an infinite straight wire. You can also assume that circuit 3 is small enough that the magnetic field is constant across it, and equal to the value at the center, a distance d from the bottom wire of circuit 2. e. The components of circuit 3 are very sensitive, and will burn out if they dissipate a power greater than Pmax. Find the inductance L for the inductor in circuit 2 that we should choose so that the components in circuit 3 do not burn out when the switch is closed. Your answer should be in terms of Req, Pmax, M, and V0. Hint: You can approximate the maximum dI/dt as (If -Ii)/T from part c. In fact this turns out not to be an approximation but the exact value. 2. Faraday’s Law A closed loop is made of a U-shaped metal wire of negligible resistance and a moveable metal crossbar of resistance R. The crossbar has mass m and length L. It is initially located a distance of h0 from the other end of the loop. The loop is placed vertically in a uniform horizontal magnetic field of magnitude B0 in the direction shown in the figure above. Express all algebraic answers in the terms B0, L, m, h0, R, and fundamental constants. a. Determine the magnitude of the magnetic flux through the loop when the crossbar is in the position shown. The crossbar is released from rest and slides with negligible friction down the U-shaped wire without losing electrical contact. b. Which direction will the current go if the crossbar falls? Justify your answer. c. Calculate the magnitude of the current in the crossbar as it falls as a function of the crossbar’s speed v. d. Derive, but do not solve, the differential equation that could be used to determine the speed v of the crossbar as a function of time. 3. Conductors, Gauss’ Law, Potential The diagram below shows a conducting cylinder with a spherical hole containing a smaller spherical conductor inside of it. We place a charge -Q on the cylinder, and a charge +Q on the sphere. a. What is the charge on the surface of the sphere, inner surface of the hole, and outer surface of the cylinder? b. What is the electric field everywhere? c. What is the potential difference between the sphere and the cylinder? 4. Superposition, Gauss’ Law A hole of radius R/2 is carved out of the inside of an insulating sphere of radius R such that the edge of the cavity is on the center of the large sphere as shown below. A positive charge .75Q is uniformly distributed over this object’s volume. a. What is the direction and magnitude of the electric field at the center of the large sphere? b. What is the direction and magnitude of the electric field at the center of the cavity? c. What is the direction and magnitude of the electric field at point P? 5. Multiloop Circuit Analysis, Kirchhoff’s Laws Two batteries, with ε1= 6.0V and ε2= 3.0V, are connected to three resistors, with 𝑅1= 6, 𝑅2= 4Ω, and 𝑅3= 2Ω, as shown below. The stars have no effect other than to denote the locations of point A and point B. a. Find the current in each resistor and the current in each battery. b. Verify that the voltage difference between point A and point B is the same no matter which path you take by evaluating the potential difference at least 3 times by taking different paths.
