DC Electricity & Capacitors

DC Electricity & DC Electricity & CapacitorsCapacitors

NCEA AS3.6NCEA AS3.6

Text Chapters 12-13Text Chapters 12-13

Kirchhoff’s LawsKirchhoff’s Laws

Current Law: The total current entering a Current Law: The total current entering a junction in a circuit equals the total current junction in a circuit equals the total current leaving.leaving.

I1=2A

I2=3A

I3=5A

321 III


Voltage Law: The total of all the potential Voltage Law: The total of all the potential differences around a closed loop in a circuit is differences around a closed loop in a circuit is zerozero

V1

V3

V2 0321 VVV


For voltage sources:For voltage sources: Going from + to – represents a loss of energy Going from + to – represents a loss of energy

so potential difference is negativeso potential difference is negative Going from – to + represents a gain in energy Going from – to + represents a gain in energy

so potential difference is positiveso potential difference is positive

+V -V


For resistors:For resistors: Passing in the same direction as the current Passing in the same direction as the current

represents a loss of energy -V=-(IR)represents a loss of energy -V=-(IR) Passing in the opposite direction to the Passing in the opposite direction to the

current represents a gain in energy +V=+(IR)current represents a gain in energy +V=+(IR)

I I

+V=+IR -V=-IR

Kirchhoff’s LawsKirchhoff’s LawsTo solve problems, To solve problems, follow a loop around a follow a loop around a circuit…circuit… a-ba-b 0V0V b-cb-c +6V+6V c-dc-d -8V (2Ax4-8V (2Ax4ΩΩ)) d-ad-a +3V (3Ax1+3V (3Ax1ΩΩ) –V) –V

Then add up the p.d’s.Then add up the p.d’s.

0V+6V-8V+3V-V=00V+6V-8V+3V-V=0

V=1V!V=1V!

V?

6V

4Ω

1Ω

a b

cd

2A

3A

Kirchhoff’s LawsKirchhoff’s LawsHarder….Harder…. a-ba-b 0V0V b-cb-c -0.75A x 4-0.75A x 4ΩΩ = -3V = -3V c-dc-d -I-I11x 2 = -2Ix 2 = -2I11VV d-ad-a +5V+5V Adding: -3 -2IAdding: -3 -2I11 +5 = 0 +5 = 0

So ISo I11 = 1A = 1A

Using the current law:Using the current law: 1A = 0.75A + I1A = 0.75A + I22

So ISo I2 2 = 0.25A= 0.25A

4Ω

2Ω

8Ω

5V

V?

0.75A

I1?

I2?

a

b c

d

e f


Continued…Continued… b-cb-c -0.75A x 4-0.75A x 4ΩΩ = -3V = -3V c-fc-f +0.25A x 8+0.25A x 8ΩΩ = +2V = +2V f-ef-e +V?+V? e-be-b 0V0V Adding: -3V + 2V + V? = 0Adding: -3V + 2V + V? = 0 So V = 1VSo V = 1V

4Ω

2Ω

8Ω

5V

V?

0.75A

I1?

I2?

a

b c

d

e f

Internal ResistanceInternal Resistance

All components have some resistance, All components have some resistance, including cells and metersincluding cells and metersThe potential difference (voltage) The potential difference (voltage) measured across a cell when no current is measured across a cell when no current is being drawn is called the E.M.F (electro-being drawn is called the E.M.F (electro-motive force)motive force)When current flows, the voltage measured When current flows, the voltage measured across the terminals of the cell will drop, across the terminals of the cell will drop, because of the cell’s internal resistance. because of the cell’s internal resistance.

Internal ResistanceInternal Resistance

A V

EMF

r

I (A)

V (V)

rEMF

IrEMFV

Electrical MetersElectrical Meters

Galvanometer – very sensitive meter that Galvanometer – very sensitive meter that can be adapted to read either current or can be adapted to read either current or voltagevoltage

Must know 2 things about the Must know 2 things about the galvanometergalvanometer The internal resistanceThe internal resistance The current that causes full scale deflection of The current that causes full scale deflection of

the needle Ithe needle If.s.df.s.d


To make a voltmeter the galvanometer To make a voltmeter the galvanometer must be connected in series with a large must be connected in series with a large resistor Rresistor Rss

G RsrSmall current passes through G

Most current continues through circuit

Electrical MetersElectrical MetersExample: Need to measure up to 10V with a Example: Need to measure up to 10V with a meter that has 200meter that has 200ΩΩ of internal resistance and of internal resistance and a full scale deflection at 5mA. a full scale deflection at 5mA. V=IRV=IR 10V= 0.005A x (200 + R10V= 0.005A x (200 + Rss)) RRss=1800=1800ΩΩ

G Rs200Ω

10V

5mA


To make an ammeter the galvanometer To make an ammeter the galvanometer must be connected in parallel with a very must be connected in parallel with a very small resistor Rsmall resistor Rpp

Small current passes through G

G

Rp

r

Most current continues through circuit

Electrical MetersElectrical MetersExample: Need to measure up to 1A with a Example: Need to measure up to 1A with a meter that has 200meter that has 200ΩΩ of internal resistance and of internal resistance and a full scale deflection at 5mA. a full scale deflection at 5mA. V (across meter) =IR = 0.005 x 200 = 1VV (across meter) =IR = 0.005 x 200 = 1V Current through RCurrent through Rpp = 1A - 0.005A = 0.995A = 1A - 0.005A = 0.995A RRpp = V/I = 1V / 0.995A = 1 = V/I = 1V / 0.995A = 1ΩΩ

G

Rp

200Ω

5mA

CapacitorsCapacitors

A capacitor is a device that can store A capacitor is a device that can store electric charge and release it later on. electric charge and release it later on.

They consist of two metal plates, They consist of two metal plates, separated by an insulating material called separated by an insulating material called a a dielectricdielectric..

Capacitors are used forCapacitors are used for Tuning circuits eg radiosTuning circuits eg radios Flashing circuits eg camera flashesFlashing circuits eg camera flashes Controlling alternating currentControlling alternating current


There are several types of capacitor:There are several types of capacitor:

Non-polarNon-polar

ElectrolyticElectrolytic

VariableVariable


Capacitors are charged by Capacitors are charged by connecting them to a power connecting them to a power supply or batterysupply or battery

This causes an accumulation This causes an accumulation of electrons on one plate of electrons on one plate (negative) and removal of an (negative) and removal of an equal number of electrons equal number of electrons from the other (positive)from the other (positive)


When a capacitor is fully charged:When a capacitor is fully charged: the current flow in the circuit stopsthe current flow in the circuit stops both plates have equal but opposite amount both plates have equal but opposite amount

of charge on themof charge on them The voltage across the plates equals the The voltage across the plates equals the

supply voltagesupply voltage An electric field exists between the plates. An electric field exists between the plates.

The strength of this field is given by: E=V/dThe strength of this field is given by: E=V/d

Capacitance CCapacitance C

The value of a capacitor is The value of a capacitor is a measure of how much a measure of how much charge it can store per volt charge it can store per volt applied across the platesapplied across the plates

The unit for capacitance is The unit for capacitance is the the farad Ffarad F

1F=1CV1F=1CV-1-1

(Most capacitors are much less (Most capacitors are much less than 1F in size, it’s a big unit!)than 1F in size, it’s a big unit!)

V

QC

CapacitanceCapacitance

The capacitance of a capacitor depends on 3 The capacitance of a capacitor depends on 3 things:things: The area of the plates which overlap (CThe area of the plates which overlap (Cαα A A

ie. The more area overlapping, the bigger ie. The more area overlapping, the bigger the C)the C)

The distance separating the plates (C The distance separating the plates (C αα 1/d 1/d ie. The closer the plates the bigger the C)ie. The closer the plates the bigger the C)

What is used as the dielectric material. What is used as the dielectric material.

CapacitanceCapacitanceThese are summed These are summed up in the capacitor up in the capacitor construction formula:construction formula:

AA = Area of overlapping plates= Area of overlapping platesdd = Distance separating plates= Distance separating plates

εεo o = Absolute permittivity of free = Absolute permittivity of free space (!!) – A constantspace (!!) – A constant= 8.84x10= 8.84x10-12-12 Fm Fm-1-1

εεrr =Dielectric constant (no units, =Dielectric constant (no units, it’s a factor)it’s a factor)

AirAir 11

Oiled paperOiled paper 22

PolystyrenePolystyrene 2.52.5

GlassGlass 66

WaterWater 8080

Charging CapacitorsCharging CapacitorsWhen the switch is When the switch is closed, charge begins closed, charge begins to build up on the to build up on the plates.plates.

The process The process continues until the continues until the cap has the same cap has the same voltage across it as voltage across it as the power sourcethe power source

Charging capacitorsCharging capacitors

It’s easy at first It’s easy at first because the plates because the plates are empty, but as are empty, but as they fill the stored they fill the stored charge starts to charge starts to repel further charge repel further charge and the rate and the rate decreasesdecreases

V

t

Vmax

Charging CapacitorsCharging Capacitors

As the rate of As the rate of charge movement charge movement decreases, the decreases, the current flow in the current flow in the circuit decreasescircuit decreases

I

t

Imax

Discharging CapacitorsDischarging Capacitors

The voltage will be The voltage will be at a max to begin at a max to begin with but will drop to with but will drop to zero as charge zero as charge leaves the plates.leaves the plates.

V

t

Vmax

Discharging CapacitorsDischarging Capacitors

The current will be The current will be large at first, but large at first, but decrease as decrease as potential left to potential left to “push” decreases“push” decreases

I

t

Imax

Charge/Discharge RatesCharge/Discharge Rates

The rate that a cap will charge or The rate that a cap will charge or discharge depends on two things:discharge depends on two things: The size of the capacitor. The bigger the cap, The size of the capacitor. The bigger the cap,

the more charge it can store, so it will take the more charge it can store, so it will take longer to fill/empty.longer to fill/empty.

The initial current in the circuit. This is The initial current in the circuit. This is determined by the resistance of the circuit. determined by the resistance of the circuit. More R means smaller current, so longer to More R means smaller current, so longer to fill/empty.fill/empty.

Time ConstantTime Constant

This is defined as the time it takes the This is defined as the time it takes the voltage/current to rise to 63% of it’s max value or voltage/current to rise to 63% of it’s max value or for it to drop 63% from it’s max valuefor it to drop 63% from it’s max value

It is calculated using this formula:It is calculated using this formula:

A cap is considered fully charged or discharged A cap is considered fully charged or discharged after after 3 3 time constants time constants

RC

Capacitor in SeriesCapacitor in Series

Joining capacitors in Joining capacitors in series:series:

21

111

CCCT

Why?Why?

The charge on each The charge on each cap is the same.cap is the same.

The voltage across The voltage across each cap adds to the each cap adds to the supply voltage supply voltage V=VV=V11+V+V22

21

21

21

111

1

1

1

CCC

Q

V

Q

V

C

Q

VV

C

Q

V

C

V

QC

V

V2

V1

+Q

+Q

-Q

-Q

Capacitor NetworksCapacitor Networks

Joining capacitors in Joining capacitors in parallel:parallel: 21 CCCT

Why?Why?

The voltage across each cap The voltage across each cap is the same as the supply.is the same as the supply.

The total charge stored is the The total charge stored is the sum of the charge in each sum of the charge in each cap. Q=Qcap. Q=Q11+ Q+ Q22

21

21

21

CCCV

Q

V

QC

V

QQC

V

QC

V

Q2Q1

(NB. This is the opposite way round to resistors)

Energy in CapacitorsEnergy in Capacitors

Capacitors store energy as electric Capacitors store energy as electric potential energy.potential energy.The amount they store is half of the The amount they store is half of the energy supplied by the battery. (The energy supplied by the battery. (The other half is dissipated as heat in the other half is dissipated as heat in the resistance of the circuit)resistance of the circuit)When a capacitor discharges, the When a capacitor discharges, the energy is dissipated in the resistance energy is dissipated in the resistance of the circuit as heat, light etc.of the circuit as heat, light etc.


For a cell: Each For a cell: Each coulomb of charge coulomb of charge gains V Joules of gains V Joules of energy energy

So the energy So the energy provided by the cell provided by the cell ΔΔE=VQE=VQ

V

Q

ΔE= area

=VQ


For a capacitor: For a capacitor: voltage increases voltage increases gradually as cap gradually as cap charges until it charges until it equals the cell equals the cell voltagevoltageSo the energy stored So the energy stored by the capacitor by the capacitor ΔΔE=1/2VQE=1/2VQ

V

Q

ΔE= area

= ½ VQ

Energy in CapacitorsEnergy in CapacitorsSome rearrangement and Some rearrangement and substitution puts this substitution puts this equation into a familiar equation into a familiar form…form…c.f. the formula for kinetic c.f. the formula for kinetic energy E=1/2mvenergy E=1/2mv22

c.f. the formula for elastic c.f. the formula for elastic potential energy E=1/2kxpotential energy E=1/2kx22

2

2

1

2

1

CVE

soV

QC

but

QVE

p

p

DC Electricity & Capacitors

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