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Lecture 22: WED 14 OCT Exam 2: Review Session CH 2427.3 & HW 0507 Physics2113 Jonathan Dowling Some links on exam stress: http://appl003.lsu.edu/slas/cas.nsf/$Content/Stress+Management+Tip+1 http://wso.williams.edu/orgs/peerh/stress/exams.html http://www.thecalmzone.net/Home/ExamStress.php http://www.staithes.demon.co.uk/exams.html Slide 2 Exam 2 (Ch24) Sec.11 (Electric Potential Energy of a System of Point Charges); Sec.12 (Potential of Charged Isolated Conductor) (Ch 25) Capacitors: capacitance and capacitors; caps in parallel and in series, dielectrics; energy, field and potential in capacitors. (Ch 26) Current and Resistance. Current Density. Ohms Law. Power in a Resistor. (Ch 27) Single-Loop Circuits, Series & Parallel Circuits, Multi-loop Circuits. NO RC Circuits. Slide 3 Electric potential: What is the potential produced by a system of charges? (Several point charges, or a continuous distribution) Electric field lines, equipotential surfaces: lines go from +ve to ve charges; lines are perpendicular to equipotentials; lines (and equipotentials) never cross each other Electric potential, work and potential energy: work to bring a charge somewhere is W = qV (signs!). Potential energy of a system = negative work done to build it. Conductors: field and potential inside conductors, and on the surface. Shell theorem: systems with spherical symmetry can be thought of as a single point charge (but how much charge?) Symmetry, and infinite systems. Slide 4 Electric potential, electric potential energy, work In Fig. 25-39, point P is at the center of the rectangle. With V = 0 at infinity, what is the net electric potential in terms of q/d at P due to the six charged particles? Slide 5 Derive an expression in terms of q 2 /a for the work required to set up the four-charge configuration of Fig. 25-50, assuming the charges are initially infinitely far apart. The electric potential at points in an xy plane is given by V = (2.0 V/m 2 )x 2 - (4.0 V/m 2 )y 2. What are the magnitude and direction of the electric field at point (3.0 m, 3.0 m)? Slide 6 Potential Energy of A System of Charges 4 point charges (each +Q) are connected by strings, forming a square of side L If all four strings suddenly snap, what is the kinetic energy of each charge when they are very far apart? Use conservation of energy: Final kinetic energy of all four charges = initial potential energy stored = energy required to assemble the system of charges +Q Do this from scratch! Don t memorize the formula in the book! We will change the numbers!!! Slide 7 Potential Energy of A System of Charges: Solution No energy needed to bring in first charge: U 1 =0 Energy needed to bring in 2nd charge: Energy needed to bring in 3rd charge = Energy needed to bring in 4th charge = +Q Total potential energy is sum of all the individual terms shown on left hand side = So, final kinetic energy of each charge = Slide 8 Potential V of Continuous Charge Distributions Straight Line Charge: dq= dx =Q/L Curved Line Charge: dq= ds =Q/2 R Surface Charge: dq= dA =Q/ R 2 dA=2 RdR Straight Line Charge: dq= dx =Q/L Curved Line Charge: dq= ds =Q/2 R Surface Charge: dq= dA =Q/ R 2 dA=2 RdR Slide 9 Potential V of Continuous Charge Distributions Straight Line Charge: dq= dx =Q/L Straight Line Charge: dq= dx =Q/L Curved Line Charge: dq= ds =Q/2 R Curved Line Charge: dq= ds =Q/2 R Slide 10 Potential V of Continuous Charge Distributions Surface Charge: dq= dA =Q/ R 2 dA=2RdR Straight Line Charge: dq= dx =bx is given to you. Straight Line Charge: dq= dx =bx is given to you. Slide 11 Slide 12 ICPP: Consider a positive and a negative charge, freely moving in a uniform electric field. True or false? (a) Positive charge moves to points with lower potential voltage. (b) Negative charge moves to points with lower potential voltage. (c)Positive charge moves to a lower potential energy. (d)Negative charge moves to a lower potential energy. Q+Q0 +V V (a) True (b) False (c) True (d) True + + + + + + + + + Slide 13 electron Slide 14 downhill Slide 15 No vectors! Just add with sign. One over distance. Since all charges same and all distances same all potentials same. Slide 16 Slide 17 xx (a) Since x is the same, only |V| matters! |V 1 | =200, |V 2 | =220, |V 3 | =200 |E 2 | > |E 3 | = |E 1 | The bigger the voltage drop the stronger the field. (b) = 3 (c) F = qE = ma accelerate leftward Slide 18 Capacitors E = / 0 = q/A 0 E = V d q = C V C plate = 0 A/d C plate = 0 A/d C sphere = 0 ab/(b-a) Connected to Battery: V=Constant Disconnected: Q=Constant Connected to Battery: V=Constant Disconnected: Q=Constant Slide 19 Isolated Parallel Plate Capacitor: ICPP A parallel plate capacitor of capacitance C is charged using a battery. Charge = Q, potential voltage difference = V. Battery is then disconnected. If the plate separation is INCREASED, does the capacitance C: (a) Increase? (b) Remain the same? (c) Decrease? If the plate separation is INCREASED, does the Voltage V: (a) Increase? (b) Remain the same? (c) Decrease? +Q Q Q is fixed! d increases! C decreases (= 0 A/d) V=Q/C; V increases. Slide 20 Parallel Plate Capacitor & Battery: ICPP A parallel plate capacitor of capacitance C is charged using a battery. Charge = Q, potential difference = V. Plate separation is INCREASED while battery remains connected. +Q Q V is fixed constant by battery! C decreases (= 0 A/d) Q=CV; Q decreases E = / 0 = Q/ 0 A decreases Does the Electric Field Inside: (a) Increase? (b) Remain the Same? (c) Decrease? Battery does work on capacitor to maintain constant V! Slide 21 Capacitors Capacitors Q=CV In series: same charge 1/C eq = 1/C j In parallel: same voltage C eq =C j Slide 22 Capacitors in Series and in Parallel What s the equivalent capacitance? What s the charge in each capacitor? What s the potential across each capacitor? What s the charge delivered by the battery? Slide 23 Slide 24 Slide 25 Capacitors: Checkpoints, Questions Slide 26 Parallel plates: C = 0 A/d Spherical: Cylindrical: C = 2 0 L/ln(b/a)] Hooked to battery V is constant. Q=VC (a) d increases -> C decreases -> Q decreases (b) a inc -> separation d=b-a dec. -> C inc. -> Q increases (c) b increases -> separation d=b-a inc.-> C dec. -> Q decreases Slide 27 PARALLEL: V is same for all capacitors Total charge = sum of Q (a)Parallel: Voltage is same V on each but charge is q/2 on each since q/2+q/2=q. (b) Series: Charge is same q on each but voltage is V/2 on each since V/2+V/2=V. SERIES: Q is same for all capacitors Total potential difference = sum of V Slide 28 Initial values: capacitance = C; charge = Q; potential difference = V; electric field = E; Battery remains connected V is FIXED; V new = V (same) C new = C (increases) Q new = ( C)V = Q (increases). Since V new = V, E new = V/d=E (same) dielectric slab Energy stored? u= 0 E 2 /2 => u= 0 E 2 /2 = E 2 /2 increases Example: Battery Connected Voltage V is Constant but Charge Q Changes Slide 29 Initial values: capacitance = C; charge = Q; potential difference = V; electric field = E; Battery remains disconnected Q is FIXED; Q new = Q (same) C new = C (increases) V new = Q/ C new = Q/ ( C) (decreases). Since V new < V, E new = V new /d = E/ (decreases) dielectric slab Energy stored? Example: Battery Disconnected Voltage V Changes but Charge Q is Constant Slide 30 Current and Resistance i = dq/dt V = i R E = J = 0 (1+ (T T 0 )) R = L/A Junction rule Slide 31 Current and Resistance Slide 32 A cylindrical resistor of radius 5.0mm and length 2.0 cm is made of a material that has a resistivity of 3.5 x 10 -5 m. What are the (a) current density and (b) the potential difference when the energy dissipation rate in the resistor is 1.0W? Slide 33 Slide 34 Current and Resistance: Checkpoints, Questions Slide 35 26.2: Electric Current, Conservation of Charge, and Direction of Current: Fill in the blanks. Think water in hose! Slide 36 All quantities defined in terms of + charge movement! (a) right (c) right(b) right (d) right Slide 37 DC Circuits Single loop Multiloop V = iR P = iV Loop rule Slide 38 Resistors vs Capacitors Resistors Capacitors Key formula: V=iR Q=CV In series: same current same charge R eq =R j 1/C eq = 1/C j In parallel: same voltage same voltage 1/R eq = 1/R j C eq =C j Slide 39 Resistors in Series and in Parallel What s the equivalent resistance? What s the current in each resistor? What s the potential across each resistor? What s the current delivered by the battery? Whats the power dissipated by each resisitor? Slide 40 Series and Parallel Slide 41 Slide 42 Slide 43 Problem: 27.P.018. [406649] Figure 27-33 shows five 5.00 resistors. (Hint: For each pair of points, imagine that a battery is connected across the pair.) Fig. 27-33 (a) Find the equivalent resistance between points F and H. (b) Find the equivalent resistance between points F and G. Slide Rules: You may bend the wires but not break them. You may slide any circuit element along a wire so long as you dont slide it past a three (or more) point junction or another circuit element. Slide 44 Slide 45 Slide 46 Too Many Batteries! Slide 47 Circuits: Checkpoints, Questions Slide 48 (a) Rightward (EMF is in direction of current) (b) All tie (no junctions so current is conserved) (c) b, then a and c tie (Voltage is highest near battery +) (d) b, then a and c tie (U=qV and assume q is +) +E+E -iR Slide 49 (a) all tie (current is the same in series) The voltage drop is iR proportional to R since i is same. Slide 50 (a) V batt < 12V (walking with current voltage drop ir (b) V batt > 12V (walking against current voltage increase +ir (c) V batt = 12V (no current and so ir=0)