Electromotive Force and Circuits

For a conductor to have a steady current, it must be a closed loop path

If charge goes around a complete circuit and returns to a starting point – potential energy does not change

As charges move through the circuit they loose their potential energy due to resistance

“Electromotive force” (emf, ε) is produced by a battery or a generator and acts as a “charge

pump”. It moves charges uphill and is equal to the potential difference across such a device under open-circuit conditions (no current). In

reality, batteries have some internal resistance.

Emf is measured in Volts (so it is not a “force” per say, but potential difference)

Sources of emf – batteries, electric generators, solar cells, fuel cells

IrV

rRI

IrIR

R

r

terminals

between Voltage

resistance Load

resistance Internal

Internal Resistance

Evolution of the electric potential

in the circuit with a load

In ideal situation, abV IR

As the charge flows through the circuit, the potentialrise as it passes through the ideal source is equal to potential drop via the resistance,

abV IR

Example: What are voltmeter and ammeter readings?

We measure currents

with ammeters

An ideal ammeter would have a zero

resistance

We measure voltages with voltmeters

An ideal voltmeter would have an infinite

resistance

Examples

Bulb B is taken away, will the bulb A glow differently?

Which bulb glows brighter?

Which bulb glows brighter?

Potential changes around the circuit

Potential gain in the battery

Potential drop at all resistances

In an old, “used-up” battery emf is nearly the same, but internal resistanceincreases enormously

Electrical energy and power

Chemical energy → Electric potential energy → Kinetic energy of charge carriers →

Dissipation/Joule heat (heating the resistor through collisions with its atoms)

R

VRIP

RIV

VIVt

Q

t

UP

QVU

Q

22

:resistor aIn

1V1A W1 :Unit

Power

pliances)devices/apin

energy of sother type(or

resistor ain heat

: chargeFor

As the charge goes through the resistance the potential energy qV is expended (if both q and V are positive), but charge does not acquire kinetic energy (current is constant). Instead, it converted to heat. The opposite can also happen – if change in potential energy is positive, the charge acquires it - battery

Power Output of a Source

rRdR

dP

rR

RRIP

0

)(

:matching) (load

load todelivered

power Maximum

2

22

2

;ab

ab

V Ir

P V I I I r

Power Input to a Source

Current flows “backwards”

2

ab

ab

V Ir

P V I I I R

Work is being done on, rather than by the top battery (source of non-electrostatic force)

Rate of conversion of electric energyinto non-electrical energy

Circuits in Series

•Resistance (light bulbs) on same path•Current has one pathway - same in every part of the circuit•Total resistance is sum of individual resistances along path•Current in circuit equal to voltage supplied divided by total resistance•Sum of voltages across each lamp equal to total voltage•One bulb burns out - circuit broken - other lamps will not light (think of

string of old Christmas lights)

ISNS 3371 - Phenomena of Nature

Water Analogy for Series Circuits

Resistors in series

)max(

Equivalent

resistorsboth in same theisCurrent

21eq

21eq

eq2121

iRRRR

RRR

IRIRIRVVV

Parallel Circuits

•Bulbs connected to same two points of electrical circuit•Voltage same across each bulb•Total current divides among the parallel branches - equals sum of current in each branch - current in each branch inversely proportional to resistance of branch•Overall resistance of circuit lowered with each additional branch•Household wiring (and new Christmas light strings) designed in parallel - too many electrical devices on - too much current - trip fuse/breaker

ISNS 3371 - Phenomena of Nature

Water Analogy for Parallel Circuits

)min(111

111 Equivalent

:junction aat splitscurrent resistors,both across same theis Voltage

21

eq

21eq

eq2121

iR

RR

R

RRR

R

V

R

V

R

VIII

Resistors in parallel

ab

hdr

ba

h

r

xr

dxdRR

drba

hdxx

h

babxr

a

b

h

2

02 )(

;)(

Calculating resistance

A variable cross-section resistor treated as a serial combination of small straight-wire resistors:

Example: Equivalent resistances

Series versus parallel connection

What about power delivered to each bulb?

2

2 2ab bc

P I R or

V VP

R R

2

2de

P I R or

VP

R

What if one bulb burns out?

Symmetry considerations to calculate equivalent resistances

No current through the resistor

I1

I1

I1

I1

I1

I1

I2

I2

I2

I2

I2

I2

rR

rIrIV

IIII

r

6

56

5)

3

1

6

1

3

1(

:b and abetween drop voltageTotal

2/ ;3/ :Currents

resistors All

121

Kirchhoff’s rules

To analyze more complex (steady-state) circuits:

1. For any junction: Sum of incoming currents equals to sum of outgoing currents

(conservation of charge)

Valid for any junction

2. For any closed circuit loop: Sum of the voltages across all elements of the loop is zero

(conservation of energy)

Valid for any close loop

- The number of independent equations will be equal to the number of unknown currents

0I

0V

Loop rule – statement that the electrostatic force is conservative.