Short Version : 25. Electric Circuits

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.

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 :

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 :

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:

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 = 

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)

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)