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Short Version : 25. Electric Circuits

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Electric Circuit = collection of electrical components connected by conductors. Examples: Man-made circuits: flashlight, …, computers. Circuits in nature: nervous systems, …, atmospheric circuit (lightning).

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25.1. Circuits, Symbols, & Electromotive Force Common circuit symbols All wires ~ perfect conductors V = const on wire Electromotive force (emf) = device that maintains fixed V across its terminals. E.g., batteries (chemical), generators (mechanical), photovoltaic cells (light), cell membranes (ions).

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Ohm’s law: Energy gained by charge transversing battery = q E ( To be dissipated as heat in external R. ) g ~ E m ~ q Lifting ~ emf Collisions ~ resistance Ideal emf : no internal energy loss.

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25.2. Series & Parallel Resistors Series resistors : I = same in every component For n resistors in series: Voltage divider Same q must go every element. n = 2 :

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Real Batteries Model of real battery = ideal emf E in series with internal resistance R int. I means V drop I R int V terminal < E

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Example 25.2. Starting a Car Your car has a 12-V battery with internal resistance 0.020 . When the starter motor is cranking, it draws 125 A. What’s the voltage across the battery terminals while starting? Voltage across battery terminals = Typical value for a good battery is 9 – 11 V. Battery terminals

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Parallel Resistors Parallel resistors : V = same in every component For n resistors in parallel :

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Analyzing Circuits Tactics: Replace each series & parallel part by their single component equivalence. Repeat.

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Example 25.3. Series & Parallel Components Find the current through the 2- resistor in the circuit. Equivalent of parallel 2.0- & 4.0- resistors: Total current is Equivalent of series 1.0- , 1.33- & 3.0- resistors: Voltage across of parallel 2.0- & 4.0- resistors: Current through the 2- resistor:

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25.3. Kirchhoff’s Laws & Multiloop Circuits Kirchhoff’s loop law: V = 0around any closed loop. ( energy is conserved ) This circuit can’t be analyzed using series and parallel combinations. Kirchhoff’s node law: I = 0at any node. ( charge is conserved )

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Multiloop Circuits INTERPRET ■ Identify circuit loops and nodes. ■ Label the currents at each node, assigning a direction to each. Problem Solving Strategy: DEVELOP ■ Apply Kirchhoff ‘s node law to all but one nodes. ( I in > 0, I out < 0 ) ■ Apply Kirchhoff ‘s loop law all independent loops: Batteries: V > 0 going from to + terminal inside the battery. Resistors: V = I R going along +I. Some of the equations may be redundant.

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Example 25.4. Multiloop Circuit Find the current in R 3 in the figure below. Node A: Loop 1: Loop 2:

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Application: Cell Membrane Hodgkin-Huxley (1952) circuit model of cell membrane (Nobel prize, 1963): Electrochemical effects Resistance of cell membranes Membrane potential Time dependent effects

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25.4. Electrical Measurements A voltmeter measures potential difference between its two terminals. Ideal voltmeter: no current drawn from circuit R m =

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Example 25.5. Two Voltmeters You want to measure the voltage across the 40- resistor. What readings would an ideal voltmeter give? What readings would a voltmeter with a resistance of 1000 give? (b) (a)

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Ammeters An ammeter measures the current flowing through itself. Ideal voltmeter: no voltage drop across it R m = 0

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Ohmmeters & Multimeters An ohmmeter measures the resistance of a component. ( Done by an ammeter in series with a known voltage. ) Multimeter: combined volt-, am-, ohm- meter.

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25.5. Capacitors in Circuits Voltage across a capacitor cannot change instantaneously.

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The RC Circuit: Charging C initially uncharged V C = 0 Switch closes at t = 0. V R (t = 0) = E I (t = 0) = E / R C charging: V C V R I Charging stops when I = 0. V R but rate I but rate V C but rate

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Time constant = RC V C ~ 2/3 E I ~ 1/3 E/R

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The RC Circuit: Discharging C initially charged to V C = V 0 Switch closes at t = 0. V R = V C = V I 0 = V 0 / R C discharging: V C V R I Disharging stops when I = V = 0.

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Example 25.6. Camera Flash A camera flash gets its energy from a 150- F capacitor & requires 170 V to fire. If the capacitor is charged by a 200-V source through an 18-k resistor, how long must the photographer wait between flashes? Assume the capacitor is fully charged at each flash.

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RC Circuits: Long- & Short- Term Behavior For t << RC: V C const, C replaced by short circuit if uncharged. C replaced by battery if charged. For t >> RC: I C 0, C replaced by open circuit.

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Example 25.7. Long & Short Times The capacitor in figure is initially uncharged. Find the current through R 1 (a) the instant the switch is closed and (b) a long time after the switch is closed. (a) (b)

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