Around the same time that the science of gases and thermodynamics was directly driving the development of 18th and 19th century industry, scientists were also developing the theories behind the industry of the 20th century, the electrical age. Although what we now call the science of electricity came from ancient observations, and was organised on a more scientific basis by people like Benjamin Franklin (1706 - 1790), Alessandro Volta (1745 - 1827) and Michael Faraday (1791 - 1867). A full and modern explanation of electricity and magnetism would not come until the work of James Clerk Maxwell (1831 - 1879) and his scientific successors, but in this section we will focus on the science and understanding developed by those earlier pioneers. Electrical fields and induction are sections in topic D. Fields.
Here we introduce the concept of charge, how charged particles can flow through conductive materials and the consequences of those currents. We will look at quantifying the electrical properties of materials and the consequences of resistance in conductors.
Electrical charge (symbol q, not to be confused with heat energy from previous sections, and with a unit of coulombs, symbol C) is a property of all materials. All objects and materials down to subatomic particles can be classified as positive (+), negative (−) or neutral, concepts first developed and named by Benjamin Franklin. Electrical charges exert forces on each other, a consequence of the existence of electromagnetic force. Like charges (+ and +, or − and −) repel each other, unlike charges (+ and −) attract each other. Neutral charges are unaffected by + or −.
A flow of charges is called an electrical current, and current (symbol I) is also a quantity of a flow of charges that can be measured. The unit of measurement is the ampere, or amp, symbol A). Electrical current is required for electrical energy to be transferred.
The full requirements for a purely electrical system of transferring energy are a source of electrical energy (measured in joules, J), a load that uses the energy and a continuous circuit of conducting material connecting them. In most examples the conducting material will be a metal wire, the load might be bulbs, motors, more general resistors or some other components. The supply could be any kind of electrical generator, but in this section we will focus on chemical and solar cells as a source of energy.
The electrical energy transferred to a circuit by a supply is called the EMF. This stands for electromotive force, but since it is not actually a force that name has been deprecated (abandoned) and we use just EMF, although some sources use source voltage. The units of EMF are joules / coulomb (J / C), given the SI derived name of volts (V). The symbol for EMF is ε, a lower-case Greek letter epsilon.
A chemical cell, also known as a battery although in science that specifically refers to two or more cells working together, supplies the EMF through chemical reactions. These separate charges inside the cell, resulting in the creation of a positive terminal with an overall positive charge and a negative terminal with an overall negative charge. Chemical cells are portable but are either single-use, which is wasteful and non-sustainable, or require recharging which requires an alternative energy source, and even rechargable cells have a limited number of recharge cycles before they become ineffective.
A solar cell also supplies EMF, using the energy of sunlight and the photovoltaic effect to separate charges and create positive and negative terminals. Unlike chemical cells, solar cells use a renewable source and are more sustainable in use, but they have less energy density (provide a smaller EMF for a given size or mass) and are only functional in a given amount of light.
As described above, a functional electrical circuit has three categories of components. The supply provides the EMF, the load is the component or components that convert electrical energy to whatever form is required (bulbs, heaters, motors, monitors, etc) and the cables that connect all of these components and allow the charges to flow and form a continuous electric current. This entire system is known as an electrical circuit and to operate must have no gaps (unless these are introduced deliberately in the form of switches to turn all or part of the circuit on or off).
The simplist type of circuit has a single path that the current can follow, connecting in series all of the components. This type of circuit is known as a series circuit. Alternatively a circuit may include junctions that allow the flow of current to split and recombine. This type of circuit is called a parallel circuit.
The standard way of showing the layout of a circuit is using a circuit diagram. Standardised symbols used in a standardised way make it easy to describe and interpret circuits. A full list of acceptable symbols is found in the Physics data booklet. The best, free, online circuit diagram tool that I am aware of is called Circuit Diagram, but be sure to use the acceptable symbols.
As well as the supply, load and cables needed for a circuit to operate we may also include the tools or meters that we use to monitor and measure the values of electrical properties in a circuit.
A supply, such as a chemical or solar cell, has two terminal one positive and one negative. Adding a conducting path between the two, for example electrical cable, will apply a force to charged particles in the conductor. In the case of metal cables, the charge particle are the lattice ions, with a positive charge, and free electrons, with a negative charge. The lattice ions are unable to more significantly, but the free electrons can, and do flow through the wire. Note that other charge carriers and mediums do exist, such as ionic solutions and semiconductors. For many electric circuit situations, in which we are concerned purely with the electric rather than the physical conditions of the circuit, the nature of the charge carriers doesn't matter. For this reason we have the concept of conventional current, which considers the charge carriers to have a positive charge and be moving from the positive terminal to the negative terminal (from high potential to low potential) even if the actual charge carriers are negative and move from negative to positive terminals.
The symbol for electrical current is the upper-case I. The units of electrical current are coulombs per second, C/s. This SI derived unit is given the name amperes, A. Amperes is often shortened to amps. From this definition we have the basic equation for electrical current:
I = ∆q/∆t
In a series circuit the current will be the same at every point in the circuit. In a parallel circuit the current may vary in different branches, but at all points - including the junctions - the current flowing into that point will equal the current leaving that point.
The standard meter we use to measure electrical current is an ammeter. Ammeters are connected in series at a chosen point in a circuit. As described above it is not very important where you connect an ammeter in a series circuit as the value should be the same everywhere, but it is significant for parallel circuits. The ammeter itself should not effect the flow of current, we say that an ideal ammeter should have zero resistance (see below for an explanation of resistance).
We have already introduced EMF is the electrical energy supplied to a circuit by a component, called the supply, with units of joules / coulomb or volts. More generally we can say that in a circuit either side of any supply (that "adds" electrical energy to a circuit) or any component of the "load" (that transforms electrical energy to other forms and purposes) there must be a difference in the electrical potential, or a potential difference. Another term for describing a potential difference is to say that there has been a change in voltage. With either term it is important to realise that there is a change or difference between two points in a circuit.
Thus, the strict definition for electrical potential (V) is the work done (W in joules), per unit charge (q in coulombs), moving a positive charge between two points along the path of the current. The equationfor this is:
V = W/q
The symbol for potential difference is V, and the units are the same as EMF, joules per coulomb, J/C, or volts, V.
The electrical meter we use for measuring electrical potential difference is the voltmeter. As the voltmeter is measuring potential difference it need to be connected in parallel between two points in a circuit and measure the potential difference, the energy transformed per unit charge, between those two points. No current should flow through the voltmeter, we say that an ideal voltmeter has no resistance (see below).
The ideal cable will conduct electricity with no losses or change to the current. No energy transformation to heat or any other form should take place in an ideal cable, and thus the electrical potential difference between any two points along a cable should be zero volts.
However, across any component of the load, there will necessarily be a potential difference, representing the energy transformed to other forms and purposes. The value of the potential difference across the component is related to the current through the component, with the ratio being determined by an electrical property of the component called the resistance, symbol R.
The definition of resistance is the ratio of the potential difference across a component (V in volts) to the current through the component (I, in amperes):
R = V / I
The unit of resistance is volts per ampere, the SI derived unit the ohm, symbol Ω, the upper-case Greek letter omega.
The word resistance is derived from thinking about how 'difficult' it is for electrical current to pass through a component. Ina high resistance component a large potential difference (a bit energy transformation) is needed to give a certain current, while in a low resistance component the same current can be produced using a smaller energy transformation, i.e. less potential difference. However, it is important to remember that the definition of resistance is the ratio of the potential difference across and the current through the component.
For a certain sample of material, the resistance of that material will depend on various properties of the material. Let us take the simple example of a component, a resistor, made from a cylinder of the material, with the electrical current entering one end of the cylinder and exiting the other end. The resistance of the resistor will be directly proportional to the length (L, in metres) of the cylinder, the longer the cylinder the greater the resistance. The resistance will also be inversely proportional to the cross-sectional area of the cylinder. The bigger this area, the smaller the resistance will be. The analogy to consider here is one of water in a pipe, the longer the pipe the more difficult it is for the water to pass through, but the wider the pipe the easier it is for the same amount of water to pass through. The other two factors to consider are the temperature of the resistor and the material itself.
Regarding temperature, for most materials, the greater the temperature, the greater the resistance. We will consider this in more detail shortly.
Regarding the material, how much resistance a material will have, for a given length and cross-sectional area (and at a given temperature) is known as the resistivity, symbol ρ, the Greek letter rho (not to be confused with the same symbol being used for density).
For a resistor of the form described above, the resistance of the resistor (at a given temperature) can be calculated using the equation: R = ρ L / A
This can be rearranged to give the equation for resistivity:
ρ = RA / L
Resistivity, ρ, has units of ohm metres. The resistivity of a material can be thought of as the resistance of a cube of that material with sides of 1 metre. For many metals this is a very small number!
Ohm's law is the idea that the current through a component is directly proportional to the potential difference across the component (I = V/R or V = IR). However, this is only valid if the resistance of the component is the same across all conditions, including temperature. For some materials, including many commercial resistors, this is approximately correct across a range of temperatures. These materials are called ohmic conductors and exhibit ohmic behaviour. Other components and materials, such as bulbs and diodes, are non-ohmic, exhibiting non-ohmic behaviour, and do not have the same resistance across a range of conditions. This is generally because of the heating effect of electrical resistors. A resistor works by converting electrical potential energy to heat. This heats up the resistor and this change in temperature changes the resistance of the resistor. Ohmic behaviour is shown in materials with a low dependence of resistance on temperature. For the purposes of this section any metal conductor at constant temperature will be considered an ohmic device..
The resistance of any component (ohmic or non-ohmic) can be found under given conditions by using the definition of resistance: measuring and calculating the ratio of potential difference across the component to the current through the component. How the resistance of a component varies with different potential differences and currents can be found by plotting a potential difference - current (V-I) graph and analysing the gradient of the graph at a point. Ohmic behaviour is shown by a constant gradient (a straight line) while a curved section of the graph shows non-ohmic behaviour.
For the purposes of this section we can normally consider all components labelled as resistors to be ohmic conductors. However, for some resistors, the value of the resistance can be changed or varied by changing some external condition and/or property of the resistor. The ones we need to consider are:
potentiometers: we vary the resistance of these variable resistors by changing the length of the conductor in the circuit.
thermistors: we vary the resistance of these variable resistors by changing the external temperature. Many thermistors are made from semiconductor materials, ceramics or polymers rather than pure metals and their resistance decreases with increasing temperatures (the opposite to how a pure metal resistor would behave).
light-dependent resistors (LDRs): the resistance of theses resistors depends on their exposure to light. In most cases the greater the exposue of the LDR to light, the lower the resistance of the LDR.
By using thermistors, LDRs and/or similar environmentally sensitive resistors we can build sensor circuits whose behaviour is dependent on the external environment.
We have previous met power, P, as the rate of doing work, W in joules, with the equation P = ∆W/∆t, where ∆t is the time taken for the work to be done in seconds. The unit of power is joules per second, J/s or watts, W. Don't confuse the symbol of watts, the unit of power, with the symbol for work done.
In electrical circuits we apply the same concept, applied a little more broadly, with the work done representing any transfer of energy.
Since potential difference is defined as joules per coulomb, and current as coulombs per second, then multiplying the two, I×V, will have units of joules per second, which is watts. This gives us the equation:
P = IV
From the equation defining resistance, R = V/I, we can substitute into the above equation to get:
P = IV = I²R = V²/R
In all cases, the units of current are amperes, A, the units or voltage are volts, V, the units or resistance are ohms, Ω, and the units of power are watts, W.
We introduced cells as electrical supplies with a given EMF, ε, in units of joules per coulomb or volts, V.
What we did not consider so far is that these cells also have an internal resistance. For most normal cells this is small, a few ohms at most, but it is not zero. The symbol for internal resistance is a lower-case r, rather the upper-case R used in the rest of the circuit. The units for internal resistance are still ohms as it is still a form of resistance.
We can find the EMF of a cell by putting a voltmeter between the terminals and reading the measurement. As an ideal voltmeter has infinite resistance (and real voltmeters have very high resistance) then no significant current should flow through the cell. Any time a current flows through a cell, as it does when all cells are operating in a functioning circuit, then potential difference across the terminals of the cell (known as the terminal voltage), which is the joules per coulomb actually supplied to the rest of the circuit, will be less than the EMF. This because of the internal resistance of the cell, turning some of the energy into heat and heating the cell, thus "wasting" the energy and making it unavailable to the rest of the circuit. This is known as the lost voltage.
It is difficult to measure the internal resistance, r, of a cell directly , and the lost voltage will depend on the current through the cell, which itself will depend on the external resistance, R, of the rest of the circuit. The internal resistance is calculated by rearranging the equation:
ε = I (R + r)
so that r = (ε / I) − R. This is normally done in an investigation by varying the external resistance using a variable resistor and plotting a suitable V-I graph.
Current: As previously described in both series and parallel circuits, the total current entering any single point in a circuit, including the junctions, always equals the total current leaving that point.
For simple series circuits, with no junctions, the implication is that the current is the same everywhere in the circuit, I = I₁ = I₂ = ....
For parallel circuits every time the circuit splits into a pair of branches, the current in both branches combined will be the original current, I = I₁ + I₂ + ... .
Potential difference: The EMF, the energy per coulomb supplied by the supply, must potential difference, joules per coulomb, transferred by the total load, all the components of the circuit, independent of the path taken by the current in the circuit.
For series circuits V = V₁ + V₂, etc. If the the potential difference across a component is 10V and the potential difference across a neighbouring component, in series, is 5V then the potential difference across both components will be 15V.
For parallel sections of a circuit, then the potential difference across both branches must be the same, V = V₁ = V₂ .This is because the same amount of energy per coulomb must be transferred by both branches. Two resistors in parallel will show the same voltage drop, even if they have difference values of resistance.
Resistance: Many circuits include many components with different resistances, R. Remember that not just resistors have resistance. Bulbs, motors and other components will have a resistance value even if that resistance changes with the conditions of the circuit. Even cells have an internal resistance, r. The effect of combining resistances arising from the relationships described above for potential difference and current.
In series circuits the resistance of resistors in series simply adds together. Two resistors of resistance R₁ and R₂ will have a combined series resistance (Rs) of R₁ + R₂. In other words, Rs = R₁ + R₂.
In parallel branches, the combined resistance of both branches (Rp) is a little more complicated to calculate. For two resistors in parallel, the inverse of the combined resistance, 1 / Rp , is given by 1 / Rp = 1 / R₁ + 1/R₂ .
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