Section 6 - Electricity - complete

- current is the rate of flow of charge
- conventional current flows from + to -
(change in) Q = I x (change in)t OR
I = (change in)Q / (change in)t
- where Q is the charge in coulombs, I is the current in amperes and t is the time taken in seconds
- the coulomb is a unit of charge
- one coulomb is defined as the amount of charge that passes in 1 second if the current is 1A
- you can measure the current flowing through a part of a circuit using an ammeter
- you would need to attach an ammeter in series so that the current through the ammeter is the same as the current through the component

- to make electric charge flow through a conductor, you need to do work on it
- potential difference, or voltage, is defined as the work done (energy converted) per unit charge moved
- V = W/Q
- you can measure the potential difference across a component using a voltmeter
- the potential difference across components in a parallel is the same, so the voltmeter should be connected in parallel with the component
- the potential difference across a component is 1 volt when you convert 1 joule of energy moving 1 coulomb of charge through the component
1V = 1JC^-1

- if you put a potential difference across an electrical component, a current will flow
- how much current you get for a particular pd depends on the resistance of the component
- mathematically, R = V/I
- where R is the resistace, V is the potential difference and I is the current
- resistance is measured in ohms
- a component has a resistance of 1 ohms is a potential difference of 1V makes a current of 1A flow through it

- a man named Ohm did most of the early work in resistance
- he developed a rule to predict how the current would change as the applies potential difference is increased, for certain types of conductors
- this rule is now called Ohm's law and the conductors that obey it (mostly metals) are called ohmic conductors

- provided the physical conditions, such as temperature, remain constant, the current through an ohmic conductor is directly proportional to the potential difference across it

- you will see a straight line graph if you plot the current against the voltage for an ohmic conductor
- doubling the pd will double the current
- this means that the resistance is constant (V/I is a fixed value)
- often factors such as light level or temperature will have a significant effect on resistace (the resistivity changes), proving that Ohm's law is only true for ohmic conductors under constant physical conditions
- ohm's law is a special case as lots of components aren't ohmic conductors and have characteristic current-voltage graphs of their own

- the term 'I/V characteristic' refers to a graph of I against V which shows how the current (I) flowing through a component changes as the potential difference (V) across it is increased
- a simple circuit would be used to obtain a characteristic I/V graph for a component
- V/I graphs are similar but with V plotted against I
- the resistance at a point on the graph is just V/I

- at a constant temperature, the current through an ohmic conductor is directly proportionl to the voltage
- this means that the I/V characteristic for an ohmic conductor at a constant temperature is a straight line through the origin
- a steep line on an I/V indicates a low resistance
- a shallow line on a V/I graph also indicates a low resistance

- the I/V characteristic for a filament lamp is a curve that starts steep but gets shallower as the voltage rises
- the filament in a lamp is just a coiled-up length of metal wire, so you might think it should have the same characteristic graph as a metallic conductor
- it doesn't because it gets hot meaning that current flowing through the lamp increases its temperature
- the resistance of a metal increases as the temperature increases

- the V/I graph for a filament lamp is a curve that starts shallow and gets steeper as the current and voltage increase

- semiconductors are not as good at conducting electricity as metals
- this is because there are far fewer charge carriers available
- however, if energy is supplied to the semiconductor, more charge carriers can be released
- this means that they make excellent sensors for detecting changes in their environment

- a thermistor is a resistor with a resistance that depends on its temperature
- NTC (Negative Temperature Coefficient) thermistors are important
- as the temperature goes up, the resistace decreases

- the I/V characteristic graph for an NTC thermistor curves upwards
- increasing the current through the thermistor increases its temperature
- the increasing gradient of this characteristic graph tells you that the resistance is decreasing

- as usual, the gradient of the V/I graph does the opposite

- warming the thermistor gives more electrons enough energy to escape from their atoms
- this means that there are more charge carrier available, so the resistance is lower

- diodes (including LEDs (light emitting diodes)) are designed to let current flow in one direction only
- forward bias is the direction in which the current is allowed to flow
- most diodes require a threshold voltage of about 0.6V in the forward direction before they will conduct
- in reverse bias, the resistance of the diode is very high and the current that flows is very tiny
- diodes let current flow in the direction that the triangle in the circuit symbol points

- the resistance of a length of wire (for example) will depend on three things:
1) Length (l) - the longer the wire, the more difficult it is to make a current flow
2) Area (A) - the wider the wire, the easier it will be for the electrons to pass along it
3) Resistivity (p) (which depends on the material) - the structure may make it easy or difficult for charge to flow
- in general, resistivity depends on environmental factors as well, like temperature and light intensity

- the resistivity of a material is defined as the resistance of a 1m length with a 1m² cross-sectional area
- it is measured in ohm-metres
- resistivity = RA / I

- typical values for the resistivity of conductors are really small, e.g. for copper (at 25^oC) resistivity = 1.72 x 10^-8 ohm-metres

- before you start, you need to know the cross-sectional area of the test wire
- assume that the wire is cylindrical, and so the cross-section is circular
- the cross sectional area would be pi x radius² 
- use a micrometer to measure the diameter of the test wire in at least three different points along the wire
- take an average value as the diameter and divide by two to get the radius (make sure it is in m)
- put it into the equation abover
- now you have your cross-sectional area

1) the test wire should be clamped to a ruler with the circuit attached to the wire where the ruler reads 0
2) attach the flying lead to the test wire
- the lead is just a wire with a crocodile clip at the end to allow connection to any point along the test wire
3) record the length of the test wire connected in the circuit, the voltmeter reading and the ammeter reading
4) use your readings to calculate the resistance of the length of the wire, using R = V/I
5) repeat this measurement and calculate an average resistance for the length
6) repeat for several different lengths, for example between 0.10 and 1.00m
7) plot the results on a graph of resistance against length and draw a line of best fit
- the gradient of the line of best fit is equal to R/I = Resistivity/A
- so multiply the gradient of the line of best fit by the cross sectional area of the wire to find the resistivity of the wire material
8) the resistivity of a material depends on its temperature, so you can only find the resistivity of a material at a certain temperature
- current flowing in the test wire can cause its temperature to increase, which can lead to random errors and invalid results
- the temperature of the wire must be constant (e.g. only have small currents flowing through the wire) 

- normally, all material have some resistivity
- that resistance means that whatever electricty flows through them, they heat up, and some of the electrical energy is wasted as thermal energy
- you can lower the resistivity of many materials like metals by cooling them down
- if you cool some materials (e.g. mercury) down to below a 'transition temperature', their resistance disappears entirely and they become a superconductor
- without any resistance, no energy is wasted 
- that means you can start a current flowing in a circuit using a magnetic field, take away the magnet and the current would carry on flowing forever
- however, most 'normal' conductors, e.g. metals, have transition temperatures below 10 Kelvin (-263^oC)
- getting things to that temperature is difficult and very expensive

- solid-state physicists are trying to develop room-temperature superconductors
- so far, they've managed to get a metal oxide to superconduct at about 140K (-133^oC)

- using superconducting wires you could make:
1 - power cables that transmit electricity without any loss of power
2 - really strong electromagnets that don't need a constant power source (for use in medical applications and Maglev trains)
3 - electronic circuits that work really fast because there's no resistance to slow them down

- power (P) is defined as the rate of transfer of energy
- it is measure in watts (W) where 1 watt is equivalent to 1 joule per second
- P = E / t

- in electrical circuits, the formula for power is:
P = V x I
- this works because:
- V is defined as the energy transferred per coulomb
- I is defined as the number of coulombs transferred per seoncd
- so V x i is energy transferred per second = power

- from the definition of resistance, it is known that V = IR
- combining the two equations gives you more ways to calculate power
- P = VI, P = V²/R and P = I² x R
- which equation you use depends on what quantities you know

- if you want to know the total energy transferred, simply multiply the power by the time:
1) E = V x I x t
2) E = (V²/R) x t
3) E = I² x R x t

Example
E = ?
V = 230V
I = 4.0 A
t = 4.5 minutes = 270 seconds

E = V x I x t
= 230 x 4 x 270
= 248400
= 250,000 J or 250 kJ

- resistance comes from electrons colliding with atoms and losing energy to other forms
- in a battery, chemical energy is used to make electrons move
- as they move, they collide with atoms inside the battery, so batteries must have resistance (internal resistance)
- internal resistance is what makes batteries and cells warm up when they are used
- chemical reactions in the battery produce electrical energy
- load resistance is the total resistance of all the components in the external circuit

- the amount of electrical energy the battery produces for each coulomb of charge is called its electromotive force (or e.m.f)
- e.m.f isnt actually a force, but it is measure in volts
- e.m.f is equal to the energy divided by the charge

- the potential difference across the load resistance is the energy transferred when one coulomb of charge flows through the load resistance
- the potential difference is called the terminal p.d (V)
- if there was no internal resistance, the terminal p.d would be the same as the e.m.f
- however, in real power supplies there is always some energy lost overcoming the internal resistance
- the energy wasted per coulomb is called the lost volts (V)

- conservation of energy tells us that:
energy per coulomb supplied by the source = energy per coulomb transferred in load resistance + energy per coulomb wasted in internal resistance

4 equations needed:
1) e.m.f = V + v 
2) V = e.m.f - v
3) e.m.f = I x (R + r)
4) V = e.m.f - Ir

where
e.m.f. = electromotive force
V = terminal p.d.
v = lost volts
I = current
R = load resistance
r = internal resistance

- for cells in series in a circuit, you can calculate the total e.m.f of the cells by adding their individual e.m.fs
- this works because each charge goes through each of the cells and so gains electromotive force from each one

- for identical cells in parallel in a circuit, the total e.m.f of the combination of cells is the same size as the e.m.f of each of the individual cells
- this is because the current will split equally between identical cells
- the charge only gains e.m.f from the cells it travels through, so the overall e.m.f in the circuit doesnt increase

1) vary the current in the circuit by changing the value of the load resistance using the variable resistor
- measure the p.d for several different values of current 

2) record the data for V and I in a table, and plot the results in a graph of V against I

3) to find the e.m.f. and internal resistance of the cell, start with the equation V = e.m.f - Ir
- rearrange to give V = -rI + e.m.f
- since e.m.f and r are constant, that's just the equation of a straight line (y = mx + c)
- so the intercept on the vertical (y) axis is e.m.f
- this also means that the gradient is equal to -r 
(gradient =  change in y / change in x )
(vertical/horizontal)

- an easier way to measure the e.m.f of a power source is by connecting a high-resistance voltmeter across its terminals
- but, a small current flows through the voltmeter so there must be some lost volts
- this means you measure a value very slightly less than the actual e.m.f but in practive the difference isnt usually significant

- as charge flows through a circuit, it doesn't get used up or lost
- this means that whatever charge flows into a junction will flow out again
- since current is rate of flow of charge, it follows that whatever current flows into a junction is the same flowing out of it 

Example
CHARGE FLOWING IN 1 SECOND
Q₁ = 6C --->  I₁ = 6A
Q₂ = 2C ---> I₂ = 2A
Q₃ = 4C ---> I₃ = 4A
I₁ = I₂ + I₃

Kirchhoff's First Law states:
"the total current entering a junction = the total current leaving it"

- energy is conserved
- in electrical circuits, energy is transferred round the circuit 
- energy transferred to a charge is e.m.f, and energy transferred from a charge is potential difference
- in a closed loop, these two quantities must be equal is energy is conserevd (which it always is)

Kirchhoff's second law says:
"the total e.m.f around a series circuit = the sum of the p.ds across each component"

Series
- same current at all points of the circuit as there are no junctions
- e.m.f split between components so: e.m.f = V₁ + V₂ + V₃
- V = IR, so if I is constant, IR_TOTAL = IR₁ + IR₂ + IR₃ 
(cancelling out the Is gives: R_total = R₁ + R₂ + R₃)

Parallel
- current is split at each junction so I = I₁ + I₂ + I₃ 
- same p.d across all components (e.g. three separate loops, within each loop the e.m.f equals sum of individual p.ds)
- so, V/Rtotal = V/R₁ + V/R₂ + V/R₃ 
- cancelling the Vs gives: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃

- at its simplest, a potential divider is a circuit with a voltage source and a couple of resistors in series
- the potential difference across the voltage source is split in the ratio of the resistances
- so, if you has a 2 ohm resistor and a 3 ohm resistor, you'd get 2/5 of the p.d across the first resistor and 3/5 of the p.d across the second one
- you can use potential dividers to supply a potential difference, V_out, between 0 and the potential difference across the voltage source
- this can be useful (for example, if you need a varying p.d or one that is lower than the voltage source)
Example
- the voltage has dropped by V₁ (the voltage across R₁) by the time it gets to this point
- the remaining voltage that can be supplied, e.g to another component, is V_out 
- in this circuit, R₂ has R₁ / (R₁ + R₂) of the total resistance, so V_out = (R₂/R₁+R₂) x V_s 
- e.g. if V_s = 9V, and you want V_out to be 6V, then you need R₂/R₁+R₂ = 2/3 which gives R₂ = 2R₁
- so you could have R₁ = 100 ohms and R₂ = 200 ohms

- this circuit is mainly used for calibrating voltmeters, which have a very high resistance
- if you put something with a relatively low resistance across R₂ though, you start to get problems
- you'll effectively have two resistors in parallel, which will always have a total resistance less than R₂ 
- that means that V_out will always be less than you've calculated and will depend on what's connected across R₂
- if you replace R₁ with a variable resistor, you can change V_out
- when R₁ = 0, V_out = V_s (as you increase R₁, V_out gets smaller)

- a light-dependent resistor has a very high resistance in the dark, but a lower resistance in the light

- an NTC thermistor has a high resistance at low temperatures, but a much lower resistance at high temperatures (it varies in the opposite way to a normal resistor)

- either of these can be used as one of the resistors in a potential divider, giving an output voltage that varies with the light level or temperature

- a potentiometer has a variable resistor replacing R₁ and R₂ of the potential divider, but it uses the same idea 

- you move a slider or turn a knob to adjust the relative sizes of R₁ and R₂ 
- that way you can vary V_out from 0V up to the source voltage

- this helps when you want to change a voltage constantly, like in the volume of the stereo