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).

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.

GOT IT? 25.1.

The figure shows three circuits.

Which two of them are electrically equivalent?

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

Example 25.1. 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?

1 2I R I R E

125

0.6

V

A

20 5 15

Voltage across lightbulb = 32 600 10 5V A 3V

Most inefficient

1

4 E

lightbulb

122.4

5.0

VA

0.6 A

1 2R RI

E

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? 25.2.

6V0V3V

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

GOT IT? 25.3.

23

3R

The figure shows all 4 possible combinations of 3 identical resistors.

Rank them in order of increasing resistance.

41

R/3 2R/33R/2

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

GOT IT? 25.4.

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 = 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 =

Conceptual Example 25.1. Measuring Voltage

What should be the electrical resistance of an ideal voltmeter?

An ideal voltmeter should not change the voltage across R2 after it is attached to the circuit.

The voltmeter is in parallel with R2.

In order to leave the combined resistance, and hence the voltage across R2 unchanged, RV must be .

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

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? 25.5.

½

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)

GOT IT? 25.6.

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