dc electricity

1

2. DC ELECTRICITY

2.1

•Charge and current

2.2

•Potential difference, e.m.f. and power

2.3

•Current-potential difference relationships

2.4

•Resistance and resistivity

2.5

•Electric circuits

2

2.1. Charge and current

Electric charge

Definition

Different units,

conversion

Electric current

Definition by its formula

Units

Number of elementary

charges flowing?

I=nAve

Understanding formula

Drift velocity

Metals, semiconducto

rs and insulators

2. DC Electricity

2.1. Charge and current 3

Electric charge (Q): Definition

Electric charge is a fundamental conserved property of electrons and protons that gives

rise to electrical forces

Charge on protons is called positive and that on electrons is called negative

All matter is made of atoms that contain protons and electrons. Charges from the

same number of electrons and protons are canceled out

Different number of protons and electrons make a body to be positively or negatively

charged


Elementary charge

unit=positive charge of an

electron=charge of a proton=e

1e=

1.6x10-19C

SI Unit of

charge=

Coulomb

(C)

Electric charge: Units

2.1 Charge and current 5

Conversions C-e


Electric current (I): Definition

I=ΔQ/ ΔtRate of flow of charge•Quanity of

charge flowing past a given point per unit time


Electric current: Units

SI current unit=Ampère=A

Flow of a charge of 1C through a given point in 1

second

1nA=10-9A

1A=1C/1s

1mA=10-3A

1μA=10-6A


Number of elementary charges flowing?

Number of e

e

Time flowing

Current passing through


Current in a material

n=number of charge

carriers per unit of

volume.It is a

property of the material. The higher

this number, the more current passes

through it.

A=cross sectional area of the sample. (usually a circle

in wires)It is not a material property.

The wider the area, the more current passes through it.

V=drift velocity of charge carriers.

It depends on the material and many other things.

e=charge passed by each carrier. (usually

electrons or

protons)I=nAve


Drift velocity This velocity is not that of the electromagnetic

wave travelling but that of the carriers in the material where the current is passing through. It is normally very slow.

Two samples of the same material but different cross sectional areas in series will have the same current passing through, so the one with the smaller area will have a higher drift velocity to compensate.

Two sample with the same section but from different material in series will have the same product “nxv”. The higher number of carriers the slower drift velocity.


Conductors, semiconductors and insulators

•n≈1028m -3 or higher •Increasing T, n remains constant, but atoms vibrate stronger and so passing current through is harder. Resistance increases.

Conductors

•n≈1019m -3•Increasi

ng T (or adding impurities), n increases very much and so current is easier to flow. Resistance decreases.

Semiconductors

•n≈0 •No current passes through them. (in normal conditions)

Insulators

DC Electricity 12

2.2 Potential difference, electromotive force and power

Introduction

Potential energies

Electrical potential energy

Potential difference and e.m.f.

Power

Electrical power

Relationships

2.2.Potential difference, electromotive force and power

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IntroductionPotential energy: Capability of a

body to do workt due to its position inside a field.

Gravitational potential energy: that which a mass has just for being near a big mass that creates a

gravitational field. (Earth) (mgΔh)

Electrical potential energy: that which a charge has just for being near a big charge that

creates an electric field. (Cell) (qEΔd)


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Electrical potential energy per charge

Dividing the energy of a

charge by its charge:

• Potentialdifference (p.d.)

• Electromotive force (e.m.f.)

Volt (V) • SI Unit

J=CV• Relationship

between units


15

Potential difference and electromotive force

Electrical energy

Other forms of energy

Potential difference

Electromotive force


16

Electrical power

• From mechanics

• Work≡EnergyP=W/Δt

• Electrical Energy

E=QV • P=VQ/Δt• Q/Δt=I

P=VI


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Relationships between P, E, V, I, Q and Δt

P=VI E=QV P=E/Δt I=Q/Δt


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PowerP=VI

P=EI/Q

P=E/Δt

P=VQ/ΔtQ/Δt=I

E=VQ I/Q=1/Δt

V=E/Q


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Electrical energy

E=QV

E=QP/I

E=PΔtE=VIΔt

V=P/I

Q/I=ΔtP=VI

IΔt=Q


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Potential difference

V=E/Q

V=E/(IΔt)

V=P/I

V=PΔt/QQ=IΔt

1/I=Δt/Q E=PΔt

PΔt=E


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ChargeQ=IΔt

Q=PΔt/VQ=E/V

Q=IE/PI=P/V

PΔt=E1/V=I/P

E/P=Δt


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Electrical current

I=Q/ΔtI=E/(VΔt)

I=P/V

I=PQ/EQ=E/V

E/Δt=P1/V=Q/E

P/E=1/Δt


23

Time of flowΔt=Q/I

Δt=E/(VI)Δt=E/P

Δt=QV/PQ=E/V

VI=PE=QV

V/P=1/I

DC Electricity 24

2.3. Current-potential difference relationships

Introduction

Resistance

Ohm’s law

I-V graphs

Ohmic materials

Tungsten filament lamps

Semiconductors diodes

2.3. I-V Relationships 25

Introduction

ResistanceThe

resistance of an electrical component is a measure of its opposition to an electric

current flowing in it

It can be defined as the

quotient between the

potential difference

across it and the current through it

R=V/IUnit “Ohm”

(Ω)Ω=VA-1


Ohm’s lawUnder certain conditions,

current is proportional to the potential difference

V/I is constant, so R is constant

Components that obey Ohm’s law are called “ohmic

components”

Ohm’s law≠Resistance


I-V graphs I-V graphs show how current behaves when

voltage varies. Knowing that “V=IR” we can realise that

“I/V=1/R” The “gradient”(slope) for a point in these

graphs shows the value for “1/R” at its certain conditions.

The bigger the gradient the lower resistance.

The smaller the gradient the higher resistance.


I-V graphs for different components

Ohmic components:

ResistorsConductors

(under certain conditions)

Tungsten filament lamps

Semiconductors diodes


Ohmic components

Conductors. (In normal conditions)

• Constant slope means constant resistance.


Filament lamps

Conductors. • For values of V close to zero, normal

conditions. So, (almost)constant slope and (almost)constant resistance.

• For values of V far from zero, T increases and resistance increases (as we saw from I=nAve). Then slope gets smaller


Diodes

Semiconductors. (under special configuration)

• For negative values of V, no current is allowed to pass through, so I=0.

• For positive values of V, (as semiconductor has impurities) n and so resistance decreases. Hence the gradient increases


Converting I-V graphs to V-I graphs

DC Electricity 33

2.4.Resistance and resistivity

Resistance

Resistor

Power dissipated

by a resistor

Resistivity

Material property

Units

2.4. Resistance and resistivity 34

Resistor• Every electrical

component, no matter its function, has a certain resistance. (R=V/I)

• Those components that have no other purpose than make current harder to pass through it are called “resistors”.Resistors are usually symbolised by…


Power dissipated by a component

P=VI

P=I2RP=V2/R

V=IRI=V/R

I2=V2/R2

Every component in a circuit will dissipate power (energy

per unit time)

This power can be shown in these 3 different ways

For a resistor, the more power it dissipates the higher

temperature it gets

For a lamp, the more power dissipated the brighter it

shines


Resistance of a sample I

Two wires of the same material

Wire1 has length l and a cross section area A

Wire2 has length l2>l and cross section area A

Wire1 (Resistance=R1)


R2>R1

R∝l The longer the resistor, the higher resistance


Resistance of a sample II

Two wires of the same material


Wire2 has length l and cross section area A2>A



R2<R1 R∝1/A The wider the resistor, the lower resistance


Resistance of a sample III

Two wires of different materials


Wire2 has length l and cross section area A



R2≠R1

R depends

on material


ResistivityR=(constant)x(l/A)

This constant is a property of the material and is called resistivity (ρ)

ρ=RA/l so units for resistivity ρ≡Ωm

Resistivity is different for each material (bigger for insulators, smaller for conductors)

Resistivity in a material changes depending on

conditions


Changes in resistivity due to temperature

Conductor Semiconductor

n constant n

A constant A constant

v ↓ v ↓

e constant e constant

I ↓ I ↑

Increasing temperature, terms in I=nAve are affected as follows:

Exponential increase of charge carriers

Because of the lattice vibrations Because of the lattice vibrations

I∝v I∝nvSo R↑ So R↓

In a conductor, resistivity increases if temperature increasesConductors have a positive temperature coefficientIn a semiconductor, resistivity decreases if temperature increasesSemiconductors have a negative temperature coefficient (NTC)NTC resistors are called thermistors. Symbolised by


Resistance(R)-Temperature(𝜃) graphs

Thermistor

𝜃

RConductor Resistor

𝜃

R

Positive Temperature Coefficient(PTC)

Negative Temperature Coefficient(NTC)

The great dependence with temperature of thermistors makes them very useful as temperature sensors

DC Electricity 42

2.5. Electric circuitsCircuit

components

Components table

Conservation of energy and

charge in circuits

Cells

Internal resistance

Solar cells

Applications

Using voltmeters

and ammeters

Resistors configurations

Potential divider

2.5. Electric circuits 43

Electric Components Cell, battery

(longer=positive)

Resistor(Conductor)

Diode(semiconductor)

Variable resistor(can vary from 0 to

R)

Thermistor(NTC)

Filament Lamp(dissipates much

heat)

Ammeter(R≈0)

Voltmeter(R≈∞)

In every component will be a potential difference fall when current passes through it


Conservation of energy in circuits

Energy dissipated by component 3



Electrical energy

converted in the cell

This means, in terms of energy per unit charge:

ε=𝚺Vi=𝚺IriElectromotive force of the cell=sum of potential difference in every component

Component 1 Component 2 Component 3

Cell


Conservation of charge in circuits

For every point in a circuit, the current “in” must be the same as the current “out”

AIi→ Io→ For point A Ii=IoFor every single wire, the current passing through it will be constant

A BI1→

I3→

I2→I1→ For point A I1=I2+I3

For point B I2+I3=I1

VA-VB is the same for both wires


Ideal and real cells An ideal cell would give all the electric

energy converted without dissipate any at all

This is, in fact, not possible. Real cells loose energy by themselves

The real value of the potential difference given by a cell is shown as: V=ε-Ir

Where ε is the electromotive force given by the cell, I is the current passing through it and r is known as internal resistance


“Measuring” internal resistance I

Connecting the cell directly to a voltmeter will give a reading of ε because the high resistance of the voltmeter will make I≈0 and then V≈ε

V=ε-Ir

Connecting the cell in series with a resistor will give a reading different to ε due to the presence of a current flowing through the internal resistance. A reading≠0


“Measuring” internal resistance II

Repeating the readings for several different resistances (a variable resistance can be used to do this) will allow us to get a V-I graph that will look like:

I

V

ε

As the straight line must obey the equation V=ε-Ir, the y intercept will show ε and the gradient will show -r


Cells efficiency Efficiency (E) of a component is defined as

the quotient between the energy given by it and the energy needed to give it

For a cell we can express this as E=V/ε But we saw before that V decreases in a

cell when current increases Efficiency of a cell decreases when current

increases To get a good efficiency of a cell we must

take care about “r” of the cell being very lower than the resistance in the circuit (so the effect of “r” will be very small)


Solar cells These are a special kind of cells, also

called photovoltaic, made of a semiconductor material

They work following the photoelectric effect

e.m.f. of these cells are constant (as normal cells)

Internal resistance in solar cells is only constant for low intensities

For larger intensities, internal resistance increases significally


Measuring current and voltage

In order to measure the current passing through a component an ammeter has to be connected in series with itTo measure the voltage fall in a component we should connect a voltmeter in parallel with it

•As resistance of the ammeter is very close to zero, does not affect the current passing through and does not dissipate energy •As resistance in the voltmeter is very large (≈∞), it does not affect current passing through the resistor as it is in parallel with an infinite resistanceReadings in A and V in

this configuration will show the current and voltage of the resistor


Resistors configurationsResistors in series:

R1 R2

is equivalent to

R=R1+R2

Resistors in parallel:

R1

R2

R=R1xR2/(R1+R2)

is equivalent to

For combinations of more than 2 resistors, just join two of them in a equivalent one and go on step by step


Potential dividerFrom conservation of energy in circuits we know that referring to the next configuration:

VA-B VB-C VC-D V

If we connect wires A and B (or any other pair) to another circuit, in fact, we are performing a new cell with a lower potential differenceDifferent values for the resistors will arrange different voltage values (always lower than V) An “ideal” potential divider will allow us to take any

possible voltage value between 0 and V This kind of potential divider can be done with a variable resistor


Variable resistors

There are two kinds of variable

resistors

Rheostats:Resistor can be changed but it has only two

terminals

Potentiometers:Three-terminal

variable resistor


Potential divider with a potentiometer

V

0≤VOUT≤V

This is the configuration for a potentiometer that allow to obtain all the possible voltage values by a potential dividerConnecting any component to VOUT we can vary the voltage on it in all the range

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