Electricity revision

A document summarising the important parts of the electricity topic.

Current and charge

Definition of current

Current is the rate of flow of electric charge.

Equation linking charge, current and time

Q = I t

Symbol	Quantity	Unit	Unit symbol
Q	Electric charge	Coulomb	C
I	Electric current	Ampere	A
t	Time	Second	s

Equation linking current, potential difference and resistance

V = I R

Symbol	Quantity	Unit	Unit symbol
I	Electric current	Ampere	A
V	Potential difference	Volt	V
R	Resistance	Ohm	Ω

Coulomb

1 coulomb is the charge that flows past a point in 1 second when there is a current of 1 amp. It’s essentially an ‘amp-second’

Calculating the number of electrons

If we know the charge or total number of electrons, we can calculate the other:

Q = n e

Symbol	Quantity	Unit	Unit symbol
Q	Electric charge	Coulomb	C
n	Number of electrons	(unitless)
e	Charge of electron	Coulomb	C

There are 6.25 \times 10^{18} electrons in 1 coulomb of charge.

Current and potential difference in series circuits

In a series circuit:

• The current is the same everywhere.
• The potential difference is shared between the components.

We can explain this using Kirchoff’s second law

Current and potential difference in parallel circuits

In a parallel circuit:

• The current is shared between the branches.
• The potential difference is the same across all branches.

We can explain this using Kirchoff’s first law

Voltage

Definition of potential difference

Potential difference is the energy transferred by 1 coulomb of charge, across a component.

Definition of electromotive force

The EMF is the energy transferred to 1 coulomb of charge, by a power source (i.e. a cell or a battery).

It’s essentially the voltage provided by the power source.

Equation linking energy, charge and potential difference

E = Q V

or:

W= Q V

Symbol	Quantity	Unit	Unit symbol
E/W	Energy transferred	Joule	J
Q	Electric charge	Coulomb	C
V	Potential difference	Volt	V

Energy transferred is sometimes represented by E, and sometimes by W (for work done, it is the same thing).

Series and parallel circuits

(see above, under ‘current and potential difference’)

Resistance

Ohm’s law

The current through a conductor at a constant temperature is directly proportional to the potential difference across the conductor.

Equation linking voltage, current and resistance

V = I R

Symbol	Quantity	Unit	Unit symbol
I	Electric current	Ampere	A
V	Potential difference	Volt	V
R	Resistance	Ohm	Ω

I-V graph for a fixed resistor

• A straight line through the origin
• This shows the current is directly proportional to the voltage
• That means that, the resistor obeys Ohm’s law

I-V graph for a filament lamp

• A curve that gets shallower as voltage increases
• This shows that as the voltage increases, the current increases at a decreasing rate
• This does not obey Ohm’s law
• The reason why the current doesn’t increase as much as the p.d. gets higher is because the filament gets hotter, so its resistance increases.
  • This means that less current can flow for a given p.d.

Why do hotter filaments have a higher resistance?

1. At higher temperatures, the atoms in the metal vibrate more.
2. This makes it more difficult for electrons to pass through the metal, because the metal ions collide more frequently with the electrons.
3. This means that the resistance increases.

I-V graph for a diode

• In the negative-voltage region (the left of the graph) the current is almost zero.
• That’s because the resistance is very high in the opposite direction.
• In the positive voltage region below the threshold voltage the current is also almost zero.
  • That’s because, below around 0.6V, diodes have a very high resistance.
• Above the threshold voltage, the current increases rapidly as the voltage increases.
  • That’s because, above around 0.6V, diodes have a very low resistance.

Thermistor

• The resistance of a thermister decreases as the temperature increases.
• That means that the current gets higher if the thermister gets hotter.
• This is the opposite of a metal wire.

Light dependent resistor (LDR)

• The resistance of an LDR decreases as the light intensity increases.
• This means that the current gets higher if the LDR is in brighter light.

Resistor networks

Resistors in series

The symbol for an ohm is Ω (the Greek letter omega).

When resistors are put in series, their resistances add up:

R_{total} = R_1 + R_2 + R_3 + ...

In other words, if we have two resistors in series, we just add up their resistances to get the total resistance.

Example

The total resistance of this circuit is 10Ω + 12Ω + 18Ω = 40Ω:

       +-----+      +-----+      +-----+   
    ---| 10Ω |------| 12Ω |------| 18Ω |---
       +-----+      +-----+      +-----+   

Resistors in parallel

If we put two resistors in parallel to each other, the total resistance of the circuit actually decreases.

That’s because the current has more options in which path it takes.

We can calculate the total resistance of resistors in parallel using this equation:

\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ...

Or, rearranged:

R_{total} = 1\div(\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ...)

Example

The total resistance of this circuit is 3Ω

       +-----+   
   +---| 6Ω  |---+
   |   +-----+   |
---|             |---
   |   +-----+   |
   +---| 6Ω  |---+
       +-----+   

To calculate this, we do:

R_{total} = 1\div(\frac{1}{6} + \frac{1}{6}) = 1\div(\frac{2}{6}) = 1\div(\frac{1}{3}) = 3Ω

Resistivity

Equation of resistivity

R = ρ \frac{L}{A}

Symbol	Quantity	Unit	Unit symbol
R	Resistance	Ohm	Ω
ρ	Resistivity	Ohm metre	Ω m
L	Length	Metre	m
A	Cross-sectional area	Metre squared	m²

What affects resistance?

From the equation above, we can find that:

• The longer the wire, the higher the resistance (directly proportional)
• The larger the cross-sectional area, the lower the resistance (inversely proportional)
• The material of the wire affects its resistivity, and that in turn affects its resistance.

Calculating the cross-sectional area of a wire

1. Find the diameter (e.g. using a micrometer)
2. Calculate the radius: r = \frac{d}{2}
3. Calculate the area using: A = \pi r^2

Power and energy

Definition of electrical power

Power is the rate of transfer of electrical energy.

Equation linking power, current and potential difference

P=VI

Symbol	Quantity	Unit	Unit symbol
P	Power	Watt	W
V	Potential difference	Volt	V
I	Electric current	Ampere	A

Equation linking power, current and resistance

P=I^2R

• We can derive this from the equation above and Ohm’s law (V=IR).
  • V=IR
  • P=VI
  • P=(IR)I
  • P=I^2R

Equation linking power, potential difference and resistance

P=\frac{V^2}{R}

• Again, we can derive this from the equation above and Ohm’s law (V=IR)!
  • I=\frac{V}{R}
  • P=VI
  • P=V(\frac{V}{R})
  • P=\frac{V^2}{R}

Equation linking energy, power and time

E = Pt

Symbol	Quantity	Unit	Unit symbol
E	Electrical energy	Joule	J
P	Power	Watt	W
t	Time	Second	s

EMF and internal resistance

Definition of internal resistance

The internal resistance is the resistance within a power supply.

Equation for internal resistance

V = E - I r

Symbol	Quantity	Unit	Unit symbol
V	Terminal p.d.	Volt	V
E	EMF	Volt	V
I	Current	Ampere	A
r	internal resistance	Ohm	Ω

• We can rearrange this to find EMF: E = V + I r

Finding the EMF from a graph of V against I

• Draw a line of best fit.
• The y-intercept is the EMF.

Finding the internal resistance from a graph of V against I

• Draw a line of best fit.
• The gradient is the negative internal resistance.
• So, to find the internal resistance, take the negative of the gradient.

Potential dividers

Definition of a potential divider

A potential divider si a circuit which takes an input voltage and then splits it across two or more components in a specific ratio.

In other words, it ‘divides’ the voltage into smaller voltages.

What is a potential divider made up of?

• Two resistors in series.
• They share the EMF in the ratio of the resistances.

Equation for potential dividers

\frac{V_1}{V_2} = \frac{R_1}{R_2}

Potentiometer

A potentiometer is a variable resistor which can be used like a potential divider.

It is not very efficient, but it allows us to get the full range of EMFs (from 0V to the maximum voltage).