Oxford Resources for IB Diploma Programme
Tim Kirk
Samplecontent
Current and circuits
Electric current
Electrical currents and energy
An electrical current involves the flow of charge (positive or negative). The name given to the thing that is moving and transporting the charge is the charge carrier. Often the charge carriers are electrons (e.g. a current along a metal wires), but they can also be ions (e.g. in a solution) or even the absence of electrons, i.e. “holes” (e.g. in a semiconductor). Whenever a current flows, electric energy is used up, so the charge carriers entering a device must have more electrical energy than when they leave. Remember that it is electrical energy that is used up, not kinetic energy. The charge carriers enter and leave at the same velocity (see page 78). Different sources of electrical energy have different advantages and disadvantages (see page 25).
The energy difference between two points in a circuit depends on the amount of charge being moved. If twice as much charge is moved between two points in a circuit, then the total energy difference will be twice as much. The basic electrical unit used to compare two points in a circuit is thus the energy difference per unit charge. This is called the potential difference or pd, between the points.
Current
Current is defined as the rate of flow of electrical charge. It is always given the symbol, I. The definition of current is expressed as follows: current = charge flowed time taken I = ΔQ Δt or (in calculus notation) I = dQ dt 1 ampere = 1
Electrical conduction in a metal
Whenever charges move you say that a current is flowing. A current is the name for moving charges and the path that they follow is called the circuit. Without a complete circuit, a current cannot be maintained for any length of time.
potential difference between two points = energy difference per unit charge moved = energy difference charge = work done charge
V = W q
The basic unit for potential difference is the joule per coulomb, J C 1. This is given a new name, the volt, V. 1 volt = 1 J C 1
Voltage and potential difference are different terms for the same thing. Potential difference is probably the better name to use as it reminds you that it is measuring the difference between two points.
The name potential difference is used because technically this quantity is just a measure of the difference between two points of something called electrical potential. Potential difference means the difference in potential. See page 124 for more detail.
It helps to get the language right. Potential difference across a device causes current through the device.
If a current flows in just one direction, then it is known as a direct current. A current that constantly changes direction (first one way then the other) is known as an alternating current or ac.
In SI units, the ampere is the base unit and the coulomb is a derived unit: 1 C = 1 A s
Samplecontent
Current flows THROUGH an object when there is a potential difference ACROSS the object. A battery (or power supply) creates the potential difference. By convention, currents are always represented as the flow of positive charge. Thus conventional current, as it is known, flows from positive to negative. Although currents can flow in solids, liquids and gases, in most everyday electrical circuits the currents flow through wires. In this case, the things that actually move are the negative electrons — the conduction electrons. The direction in which they move is opposite to the direction of the representation of conventional current. As they move the
interactions between the conduction electrons and the lattice ions means that work needs to be done. Therefore, when a current flows, the metal heats up. The speed of the electrons due to the current is called their drift velocity
▴ Electrical conduction in a metal conduction electrons conventional current, I metal wire positive ions held in place dri velocity
The drift speed equation is I = nAvq
The typical drift speed of an electron is 10–4 m s–1
(5 A current in metal conductor of cross section 1 mm2)
Internal resistance and cells
Electromotive force and internal resistance
When a 6 V battery is connected in a circuit some energy will be used up inside the battery itself. In other words, the battery has some internal resistance. The TOTAL energy difference per unit charge around the circuit is still 6 volts, but some of this energy is used up inside the battery. The energy difference per unit charge from one terminal of the battery to the other is less than the total made available by the chemical reaction in the battery.
For historical reasons, the TOTAL energy difference per unit charge around a circuit is called the electromotive force (emf). However, remember that it is not a force (measured in newtons) but an energy difference per charge (measured in volts).
In practical terms, emf is exactly the same as potential difference if no current flows.
Cells and batteries
An electric battery is a device consisting of one or more cells joined together. In a cell, a chemical reaction takes place, which converts stored chemical energy into electrical energy. There are two different types of cell: primary and secondary.
A primary cell cannot be recharged. During the lifetime of the cell, the chemicals in the cell get used in a non-reversible reaction. Once a primary cell is no longer able to provide electrical energy, it is thrown away. Common examples include zinc–carbon batteries and alkaline batteries.
A secondary cell is designed to be recharged. The chemical reaction that produces the electrical energy is reversible. A reverse electrical current charges the cell allowing it to be reused many times. Common examples include a lead–acid car battery, or nickel–cadmium and lithium–ion batteries.
The charge capacity of a cell is how much charge can flow before the cell stops working. Typical batteries have charge capacities that are measured in amp-hours (A h).
1 A h is the charge that flows when a current of 1 A flows for one hour, i.e. 1 A h = 3 600 C.
Determining internal resistance experimentally
The circuit below can be used to experimentally determine the internal resistance r of a cell (and its emf ε).
Samplecontent
Procedure:
• Vary external resistance R to get a number (ideally 10 or more) of matching readings of V and I over as wide a range as possible.
• Repeat readings.
• Do not leave current running for too long (especially at high values of I).
• Take care that nothing overheats.
Data analysis:
• The relevant equation, V = ε Ir was introduced above.
• A plot of V on the y-axis and I on the x-axis gives a straight line graph with ⚪ gradient = r ⚪ y-intercept = ε
Electric circuits
Ohm’s law — ohmic and non-ohmic behaviour
The graphs show how the current varies with potential difference for some typical devices.
If current and potential difference are proportional, the device is said to be ohmic (e.g. a metal at constant temperature). Devices where current and potential difference are not proportional are said to be non-ohmic (e.g. a filament lamp or a diode).
(a) metal at constant temperature (b) filament lamp (c) diode potential difference current potential difference current potential difference current
Samplecontent
Ohm’s law states that the current flowing through a piece of metal is proportional to the potential difference across it providing the temperature remains constant.
Power dissipation
Since potential difference = energy difference charge flowed and current = charge flowed time taken then that potential difference × current = energy difference charge flowed × charge flowed time taken =
Circuits — Kirchoff’s circuit laws
An electric circuit can contain many different devices or components. The mathematical relationship V = IR can be applied to any component or group of components in a circuit.
When analysing a circuit it is important to look at the circuit as a whole. The power supply is the device that is providing the energy, but it is the whole circuit that determines what current flows through the circuit. Two fundamental conservation laws apply when analysing circuits: the conservation of electric charge and the conservation of energy. These laws are collectively
Resistance
Resistance is the mathematical ratio between potential difference and current. If something has a high resistance, it means that you would need a large potential difference across it to get a current to flow.
Resistance = potential difference current
In symbols, R = V I
In symbols, V ∝ I (if temperature is constant)
A device with constant resistance (i.e. an ohmic device) is called a resistor
The symbols that are used when representing electrical circuits are all shown on page 5 of the Physics Data booklet.
This energy difference per time is the power dissipated by the resistor. All this energy is going into heating up the resistor. In symbols: P = V × I
Sometimes it is more useful to use this equation in a slightly different form, e.g. P = V × I and V = I × R, so P = I × R ()
Similarly, P = V 2 R
b) R = 250 4.8 = 52 Ω energy difference time
A new unit, the ohm, Ω, is defined to be equal to one volt per amp: 1 ohm = 1 V A 1
known as Kirchoff’s circuit laws and can be stated mathematically as:
first law: I = 0 (junction)
second law: V = 0 (loop)
The first law states that the total current going into any junction in the circuit must be the same as the current coming out of the junction.
The second law states that around any loop, the total energy per unit charge must sum to zero. Any source of potential difference within the loop must be completely dissipated across the components in the loop.
Worked example
A 1.2 kW electric kettle is plugged into the 250 V mains supply. Calculate:
a) the current drawn b) the resistance of the kettle.
a) I = 1200 250 = 4.8 A
Resistors in series and parallel
Resistors in series
A series circuit has components connected one after another in a continuous chain. The current must be the same everywhere in the circuit since charge is conserved. The total potential difference is shared among the components.
Total resistance = 3 Ω + 4 Ω + 5 Ω = 12 Ω
You can work out what share they take by looking at each component in turn, e.g. the potential difference across the resistor = I × R1 the potential difference across the bulb = I × R2
energy is converted into:
Rtotal = R1 + R2 + R3
This always applies to a series circuit. Note that V = IR correctly calculates the potential difference across each individual component as well as calculating it across the total.
Resistors in parallel
▴ Example of a series circuit
A parallel circuit branches and allows the charges to take more than one possible route around the circuit.
The total current is the sum of the currents in each branch.
▴ Example of a parallel circuit
Since the power supply fixes the potential difference, each component has the same potential difference across it.
Electrical meters
A current-measuring meter is called an ammeter. It should be connected in series at the point where the current needs to be measured. A perfect ammeter would have zero resistance. A real ammeter behaves like a perfect ammeter with a small non-negligible resistance in series with it.
A meter that measures potential difference is called a voltmeter. It should be placed in parallel with the component or components being considered. A perfect voltmeter has infinite resistance. A real voltmeter behaves like a perfect voltmeter with a large non-infinite resistance in parallel with it.
Potential divider circuits and sensors
Potential divider circuit
The circuit diagram shown in question 1 on page 82 is an example of a circuit involving a potential divider. It is so called because the two resistors “divide up” the potential difference of the battery. You can calculate the “share” taken by one resistor from the ratio of the resistances but this approach does not work unless the voltmeter’s resistance is considered. An ammeter’s internal resistance also needs to be considered. One of the most common mistakes when solving problems involving electrical circuits is to assume that the current or potential difference remains constant after a change to the circuit. After a change, the only way to ensure that your calculations are correct is to start again.
A variable potential divider (a potentiometer) is often the best way to produce a variable power supply. When designing the potential divider, the smallest resistor that is going to be connected needs to be taken into account: the potentiometer’s resistance should be significantly smaller.
A potentiometer has three terminals – the two ends and the central connection
output voltage
Sensors
A light-dependent resistor (LDR), is a device whose resistance depends on the amount of light shining on its surface. An increase in light causes a decrease in resistance. LDR
When light shines on the LDR, there will be a decrease in pd across the LDR.
To measure the V–I characteristics of an unknown resistor R, the two circuits (A and B) below are constructed. Both will both provide a range of readings for the potential difference, V, across and current, I, through R. Providing that R is much greater than the resistance of the potentiometer, circuit B is preferred because the range of readings is greater.
• Circuit B allows the potential difference across R (and hence the current through R) to be reduced down to zero. Circuit A will not go below the minimum value achieved when the variable resistor is at its maximum value.
• Circuit B allows the potential difference across R (and hence the current through R) to be increased up to the maximum value Vsupply that can be delivered by the power supply in regular intervals. The range of values obtainable by Circuit A depends on the maximum resistance of the variable resistor.
Samplecontent
When light shines on the LDR, there will be an increase in pd across the fixed resistor.
A thermistor is a resistor whose value of resistance depends on its temperature. Most are semiconducting devices that have a negative temperature coefficient (NTC). This means that an increase in temperature causes a decrease in resistance. Both of these devices can be used in potential divider circuits to create sensor circuits. The output potential difference of a sensor circuit depends on an external factor.
When the temperature of the thermistor increases, there will be an increase in pd across the fixed resistor 10kΩ
When the temperature of the thermistor increases, there will be a decrease in pd across the thermistor
Resistivity
The resistivity, ρ, of a material is defined in terms of its resistance, R, its length l and its cross-sectional area A
R = ρ l A
The units of resistivity must be ohm metres (Ω m). Note that this is the ohm multiplied by the metre, not “ohms per metre”.
Investigating resistance
The resistivity equation predicts that the resistance R of a substance will be:
• proportional to the length l of the substance
• inversely proportional to the cross-sectional area A of the substance.
These relationships can be predicted by considering resistors in series and in parallel.
• Increasing l is like putting another resistor in series. Doubling l is the same as putting an identical resistor in series. R in series with R has an overall resistance of 2R. Doubling l means doubling R So R ∝ l. A graph of R vs l will be a straight line going through the origin.
• Increasing A is like putting another resistor in parallel. Doubling A is the same as putting an identical resistor in parallel. R in parallel with R has an overall resistance of R 2
Doubling A means halving R. So R ∝ 1 A . A graph of R vs 1 A will be a straight line going through the origin.
Worked example
The resistivity of copper is 3.3 × 10 7 Ω m. Calculate the resistance of a 100 m length of wire of cross-sectional area 1.0 mm2
R = 3.3 × 10 7 × 100 10 4 = 0.33 Ω
To practically investigate these relationships, you have:
Independent variable Either l or A
Control variables A or l (depending on above choice)
Temperature
Substance.
Data collection For each value of independent variable:
• a range of values for V and I should be recorded
• R can be calculated from the gradient of a V vs I graph.
Data analysis
Resistivity Samplecontent
Values of R and the independent variable are analysed graphically.
Possible sources of error/uncertainty include:
• Temperature variation of the substance (particularly if currents are high). Circuits should not be left connected.
• The cross-sectional area of the wire is calculated by measuring the wire’s diameter, d, and using
A = πr 2 = πd 2 4 . Several sets of measurements should be taken along the length of the wire and the readings in a set should be mutually perpendicular.
• The small value of the diameter of the wire will mean that the uncertainties generated using a ruler will be large. This will be improved using a vernier caliper or a micrometer
End of topic questions — B.5 Current and circuits
1. In the circuit below, the voltmeter has a resistance of 20 kΩ. Calculate:
a. the pd across the 20 kΩ resistor with the switch open (2)
b. the reading on the voltmeter with the switch closed. (3)
2. A student connects the circuit below to take current and voltage readings to calculate the resistance of R
b. Sketch an appropriate graph to estimate the emf of the cell.
i. Estimate the emf of the cell. (3)
ii. Estimate the internal resistance of the cell. (3)
Another student attempts to repeat the first experiment, but they accidentally swap the positions of the voltmeter and ammeter to give the following incorrect circuit: V
The student records the following data for different values of the variable resistor:
The student notices that the readings on the voltmeter and ammeter do not change for different values of the variable resistor.
c. Determine the value shown on:
i. the ammeter (3)
ii. the voltmeter. (3)
3. A battery is connected to two 2.5 Ω resistors connected in parallel. The total current flowing out of the battery is 5.1 A. The resistors are then connected in series and the total current flowing out of the battery is 2.0 A. Calculate:
a. the internal resistance of the battery (3)
b. the emf of the battery (3)
c. the power dissipated in one of the resistors:
i. when connected in parallel (3)
ii. when connected in series. (2)
a. Sketch an appropriate graph to estimate the value of the unknown resistance R. (5)
The student now moves the voltmeter to be across the terminals of the cell and takes another set of data: