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PHYS152 Lecture 6 Ch 28 Circuits Eunil Won Korea University Fundamentals of Physics by Eunil Won, Korea University Work, Energy, and Emf (기전력) To produce a steady flow of charge, one need a “charge pump” : we call it an emf (electromotive force) device ex) battery, electric generator, solar cells In any time interval dt, a charge dq passes through and we define the emf as dW E= dq SI unit of emf : joule per coulomb = volt emf arrow: from negative terminal toward the positive terminal Calculating the Current in a Single-Loop Circuit dW = E dq = Ei dt 2 Ei dt = i R dt Fundamentals of Physics by Eunil Won, Korea University and this has to be equal the thermal energy that appears in the resistor so E = iR or E i= R Calculating the Current in a Single-Loop Circuit Loop Rule: The algebraic sum of the changes in potential encountered in a complete traversal of any loop of a circuit must be zero (=Kirchhoff’s loop rule) We traverse a complete loop (clockwise): Va + E − iR = Va so For a counterclockwise travel, we get: E − iR = 0 −E + iR = 0 For more complex circuits we prepare following rules: Resistance rule: For a move through a resistance in the direction of the current the change in potential is -iR (in the opposite direction it is +iR) Emf rule: For a move through an ideal emf device in the direction of the emf arrow, the change in potential is +E (in the opposite direction it is -E) Fundamentals of Physics by Eunil Won, Korea University Other Single-Loop Circuits Internal Resistance: real battery has internal resistance (r) applying the loop rule: E − ir − iR = 0 or E i= R+r Resistances in Series applying the loop rule: E − iR1 − iR2 − iR3 = 0 E i= R1 + R2 + R3 Req = R1 + R2 + R3 The extension to n resistances is (mathematical induction) Fundamentals of Physics by Eunil Won, Korea University Req = n ! j=1 Rj Potential Differences Potential difference applying the loop rule: Vb − iR = Va E from the previous result of i = R+r so we get Power, Potential, and emf Vb − Va = +iR R Vb − Va = E R+r The net rate of energy transfer (P) from the emf device to the charge carriers is: P = iV where V is the potential across the terminals of the emf device P = i(E − ir) = iE − i2 r Pemf = iE Pr = i2 r (internal dissipation rate) Fundamentals of Physics by Eunil Won, Korea University (power of emf device) Multiloop Circuits Let’s consider the junction d: charge conservations requires i1 + i3 = i2 Junction Rule: The sum of the currents entering any junction must be equal to the sum of the currents leaving that junction (Kirchhoff’s junction rule) By applying loop rule (counterclockwise) to left-hand loop By applying loop rule (counterclockwise) to right-hand loop By applying loop rule (counterclockwise) to big loop E1 − i1 R1 + i3 R3 = 0 −i3 R3 − i2 R2 − E2 = 0 E1 − i1 R1 − i2 R2 − E2 = 0 note: there are three unknown currents and we need three equations to solve them completely (The last equation is sum of two from loop rule) Resistances in Parallel V V V , i = , i = Definitions of resistance gives: 2 3 R1 R2 R3 ! " Applying junction rule at point a: 1 1 1 i = i1 + i2 + i3 = V + + R1 R2 R3 i1 = Fundamentals of Physics by Eunil Won, Korea University 1 1 1 1 = + + Req R1 R2 R3 n ! 1 1 = (n resistances in parallel) Req R j j=1 RC Circuits Charging a Capacitor : capacitor is charged when the switch is closed applying the loop rule to the circuit since i= dq dt q E − iR − =0 C q dq we get the following differential equation R + =E dt C Let’s assume that the general solution to this has the form we require so t → ∞, q = qp + Ke dq =0 dt so qp : a particular solution K : a constant to be evaluated −at qp = CE (from the initial condition) q dq (from R + = E ) dt C q = CE + Ke−at initial condition: t = 0, plugging it into the differential equation gives q=0 RCEae finally we get: Fundamentals of Physics by Eunil Won, Korea University gives −at 0 = CE + K + E − Ee −at =E so q = CE − CEe−at 1 a= RC q = CE(1 − e−t/RC ) RC Circuits q = CE(1 − e−t/RC ) The charging process is now fully described by the solution above The current charging the capacitor can be obtained as the time derivative of the charge dq i= = dt ! E R " The capacitive time constant e−t/RC (1Ω x 1 F = 1 s) τ = RC Discharging a Capacitor (no emf) Fundamentals of Physics by Eunil Won, Korea University q dq =0 R + q = q0 e−t/RC dt C ! q " dq 0 =− e−t/RC i= dt RC Summary Emf: Loop rule: Series Resistances dW E= dq The algebraic sum of the changes in potential encountered in a complete traversal of any loop of a circuit must be zero (=Kirchhoff’s loop rule) Req = n ! Rj (n resistances in series) j=1 ! 1 1 = Req Rj j=1 n Resistance RC circuits Fundamentals of Physics by Eunil Won, Korea University (n resistances in parallel) q = CE(1 − e−t/RC )