25. Electric Circuits 1.Circuits, Symbols, & Electromotive Force 2.Series & Parallel Resistors 3.Kirchhoff’s Laws & Multiloop Circuits 4.Electrical Measurements 5.Capacitors in Circuits

Festive lights decorate a city. If one of them burns out, they all go out. Are they connected in series or in parallel?

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

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).

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.

GOT IT? The figure shows three circuits. Which two of them are electrically equivalent?

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 :

Example Voltage Divider A lightbulb with resistance 5.0  is designed to operate at a current of 600 mA. To operate this lamp from a 12-V battery, what resistance should you put in series with it? Voltage across lightbulb = Most inefficient lightbulb

Rank order the voltages across the identical resistors R at the top of each circuit shown, and give the actual voltage for each. In (a) the second resistor has the same resistance R, and (b) the gap is an open circuit (infinite resistance). GOT IT? V 0V 3V

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

Example Starting a Car Your car has a 12-V battery with internal resistance . 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

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

GOT IT? R The figure shows all 4 possible combinations of 3 identical resistors. Rank them in order of increasing resistance. 41 R/32R/3 3R/2

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

Example 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- ,  & 3.0-  resistors:  Voltage across of parallel 2.0-  & 4.0-  resistors: Current through the 2-  resistor:

GOT IT? dimmer The figure shows a circuit with 3 identical lightbulbs and a battery. (a) Which, if any, of the bulbs is brightest? (b) What happens to each of the other two bulbs if you remove bulb C? brighter

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 )

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.

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

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

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

Conceptual Example Measuring Voltage What should be the electrical resistance of an ideal voltmeter? An ideal voltmeter should not change the voltage across R 2 after it is attached to the circuit. The voltmeter is in parallel with R 2. In order to leave the combined resistance, and hence the voltage across R 2 unchanged, R V must be .

Example 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)

If an ideal voltmeter is connected between points A and B in figure, what will it read? All the resistors have the same resistance R. GOT IT? ½ E

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

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.

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

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 

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

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.

Example 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.

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.

Example 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)

GOT IT? A capacitor is charged to 12 V & then connected between points A and B in the figure, with its positive plate at A. What is the current through the 2-k  resistor (a)immediately after the capacitor is connected and (b) a long time after it’s connected? 6 mA 2 mA