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).

IR

EOhm’s law:

Energy gained by charge transversing battery = q ( 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

1 2V V E 1 2I R I R sI R

1 2sR R R

For n resistors in series: 1

n

s jj

R R

s

IR

E

j jV I R j

s

R

R E

Voltage divider

Same q must go every element.

n = 2 :1

11 2

RV

R R

E 2

21 2

RV

R R

E

Real Batteries

Model of real battery = ideal emf in series with internal resistance Rint .

int LI R I R Eint L

IR R

E

LR

E

I means V drop I Rint

Vterminal <

intL

LR

L

RV

R R

E

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?

int LI R I R E

Voltage across battery terminals = int 12 125 0.020V V A E 9.5V

Typical value for a good battery is 9 – 11 V.

Battery terminals

intLR RI

E 12

0.020125

V

A

0.096 0.020 0.076

Parallel Resistors

Parallel resistors :

V = same in every component

1 2I I I 1 2R R

E E

pRE

1 2

1 1 1

pR R R

For n resistors in parallel :

1

1 1n

jp jR R

1 2

1 2p

R RR

R R

Analyzing Circuits

Tactics:

• Replace each series & parallel part by their single component equivalence.

• Repeat.

Example 25.3. Series & Parallel Components

Find the current through the 2- resistor in the circuit.

Equivalent of parallel 2.0- & 4.0- resistors:

1 1 1

2.0 4.0R

1.33R

Total current is

Equivalent of series 1.0-, 1.33- & 3.0- resistors:

5.33

3

4.0

1.0 1.33 3.0TR

5.33T

IR E 12

5.33

V

2.25 A

Voltage across of parallel 2.0- & 4.0- resistors: 1.33 2.25 1.33V A 2.99V

Current through the 2- resistor: 2

2.99

2.0

VI

1.5A

25.3. Kirchhoff’s Laws & Multiloop Circuits

Kirchhoff’s loop law:

V = 0 around any closed loop.

( energy is conserved )

This circuit can’t be analyzed using series and parallel combinations.

Kirchhoff’s node law:

I = 0 at 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. ( Iin > 0, Iout < 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 25.4. Multiloop Circuit

Find the current in R3 in the figure below.

Node A:

1 2 3 0I I I

Loop 1: 1 1 1 3 3 0I R I R E

3

1 1 91 3

2 4 4I

2 2 2 3 3 0I R I R E

1 36 2 0I I

2 39 4 0I I

1 3

13

2I I 2 3

1 9

4 4I I

3

4 21

7 4I 3A

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 Rm =

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?

40

4012

40 80V V

(b) 40 1000

40 1000parallelR

4V

38.5

(a)

40

38.512

38.5 80V V

3.95V

Ammeters

An ammeter measures the current flowing through itself.

Ideal voltmeter: no voltage drop across it Rm = 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 VC = 0

Switch closes at t = 0.

VR (t = 0) =

I (t = 0) = / R

C charging: VC VR I

Charging stops when I = 0.VR but rate I but rate

VC but rate

0Q

I RC

E

0d I I

Rd t C

dQI

d t

d I d t

I RC

0 0

I t

I

d I d t

I RC

0

lnI t

I RC

0

t

RCI I e

t

RCeR

E

C RV V E 1t

RCe

E

Time constant = RC

VC ~ 2/3

I ~ 1/3 /R

The RC Circuit: Discharging

C initially charged to VC = V0

Switch closes at t = 0.

VR = VC = V

I 0 = V0/ R

C discharging: VC VR I

Disharging stops when I = V = 0.

0Q

I RC

d I d t

I RC

dQI

d t

0

t

RCI I e

0t

RCVe

R

0

t

RCV V e

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.

ln 1 CVt RC

E

5.1 s

3 6 17018 10 150 10 ln 1

200

VF

V

RC Circuits: Long- & Short- Term Behavior

For t << RC: VC const,

C replaced by short circuit if uncharged.

C replaced by battery if charged.

For t >> RC: IC 0,

C replaced by open circuit.

Example 25.7. Long & Short Times

The capacitor in figure is initially uncharged.

Find the current through R1

(a) the instant the switch is closed and

(b) a long time after the switch is closed.

11

IR

E

(a)

11 2

IR R

E

(b)