3.2.1 Series and parallel circuits

Series circuits

1. In a series circuit, the components are joined end to end so that they form a single loop with the power supply.
2. There is only one path around the circuit, so all of the charge that leaves the supply passes through every component in turn.
3. The current is the same at every point in a series circuit, because charge is not used up by the components and cannot leave the loop anywhere.
   1. An ammeter placed before the first lamp, between two lamps or after the last lamp gives the same reading.
4. The potential difference of the supply is shared between the components, so the potential differences across the components add up to the potential difference across the supply.
   1. For two components in series, this is written as $V_{supply}=V_1+V_2$.
   2. The component with the larger resistance takes the larger share of the potential difference.
5. In terms of energy, each coulomb of charge is given energy by the supply and then transfers part of that energy in each component it passes through.
   1. The energy transferred by each coulomb in all of the components adds up to the energy it was given by the supply.
6. If one component is removed, breaks or is switched off, the single loop is broken and every component stops working.
   1. Old-style strings of festive lights were wired in series, so one failed bulb made the whole string go dark.
7. One switch placed anywhere in a series circuit turns all of the components on or off together.

Parallel circuits

1. In a parallel circuit, each component is placed on its own branch, and every branch is connected between the same two points of the circuit.
2. The current from the supply splits at a junction, part of it flows along each branch, and the branch currents recombine at a second junction.
3. The total current from the supply is equal to the sum of the branch currents, written $I_{total}=I_1+I_2$ for two branches.
   1. A branch with a smaller resistance carries a larger current than a branch with a larger resistance.
4. Each branch is connected directly across the supply, so the potential difference across every branch is the same and equal to the supply potential difference.
5. In terms of energy, each coulomb of charge passes through only one branch, so it transfers all of the energy it was given by the supply in that branch.
6. Each branch can be switched on and off independently, and a break in one branch does not stop charge flowing through the others.
   1. The lights and sockets in a house are wired in parallel, so every appliance receives the full mains potential difference and switching off the kettle does not switch off the fridge.
7. Adding another identical lamp in series shares the supply potential difference between more lamps, so each lamp gets a smaller share and glows more dimly.
8. Adding another identical lamp in parallel gives it the full supply potential difference, so it glows as brightly as the lamps already there, while the total current from the supply increases.

Comparing the two circuits

1. Paths: a series circuit has one path for the charge, whereas a parallel circuit has a separate path through each branch.
2. Current: the current is the same everywhere in a series circuit, whereas in a parallel circuit the branch currents add up to the total current.
3. Potential difference: the supply potential difference is shared between series components, whereas each parallel branch has the full supply potential difference across it.
4. Energy: each coulomb transfers some of its energy in every series component, whereas it transfers all of its energy in the one parallel branch it passes through.
5. Faults and switching: one break stops a whole series circuit, whereas a break in one parallel branch leaves the other branches working.

Positioning measuring instruments

1. An ammeter measures the current through a component in amperes, $A$.
   1. To fit an ammeter, the circuit is broken at the point of interest and the ammeter is connected in series so that the same charge flows through it.
   2. An ammeter has a very low resistance, so it does not noticeably reduce the current it is measuring.
2. In a series circuit, an ammeter can go anywhere in the loop because the current is the same at every point.
3. In a parallel circuit, an ammeter in the main part of the circuit measures the total current, while an ammeter in a branch measures only the current in that branch.
4. A voltmeter measures the potential difference across a component in volts, $V$.
   1. A voltmeter is connected in parallel with the component, with one lead on each side of it, and the circuit does not need to be broken.
   2. A voltmeter has a very high resistance, so almost no current flows through it and it does not change the circuit it is measuring.
5. A voltmeter across the supply measures the supply potential difference, which equals the sum of the series voltmeter readings or the reading across any one parallel branch.
6. The positive terminal of each meter should be connected to the side of the circuit nearest the positive terminal of the supply, or a digital meter shows a negative reading and an analogue needle moves the wrong way.

3.2.2 Circuit symbols and circuit diagrams

Direct current circuits

1. A d.c. circuit is a circuit in which the current always flows in the same direction, supplied by a cell, a battery or a d.c. power supply.
2. Every d.c. supply has two terminals, a positive terminal ($+$) and a negative terminal ($-$).
3. In a metal wire, the moving charges are electrons, which are negatively charged, so they leave the negative terminal, travel around the circuit and return to the positive terminal.
4. The direction of current was chosen before electrons were discovered, and scientists agreed to show current flowing from positive to negative.
5. This agreed direction, called conventional current, is still used on every circuit diagram.
   1. Conventional current flows out of the positive terminal, around the circuit and back into the negative terminal, which is the opposite direction to the flow of electrons.
6. The positive and negative terminals matter whenever a component only works one way round, such as a diode, or whenever a meter has its own positive and negative terminals.

Circuit diagrams

1. A circuit diagram uses standard symbols instead of pictures, so that anyone can read it in the same way whatever the real components look like.
2. Connecting wires are drawn as straight lines, usually horizontal or vertical, so that most circuit diagrams look like a set of rectangles.
3. A junction, where wires are joined, is shown by a solid dot, while two lines that cross without a dot are not connected.
4. A circuit diagram shows how the components are connected, not where they sit on the bench or how long the wires are.
   1. Connecting wires are treated as having no resistance, so a long wire on a diagram behaves exactly like a short one.
5. A complete circuit is a closed loop of conductors running from one terminal of the supply, through the components, back to the other terminal.
   1. Any gap in the loop, apart from a switch that is meant to be open, means that no current can flow.

Supplies and switches

1. A cell is drawn as two parallel lines of different lengths, where the long thin line is the positive terminal and the short thick line is the negative terminal.
2. A battery is drawn as two or more cell symbols joined together, sometimes with a dashed line between the first and last cell.
   1. When the cells all face the same way, the potential difference of the battery is the sum of the potential differences of the cells, so two $1.5V$ cells give $3.0V$.
3. A d.c. power supply is usually drawn as two small open circles marked $+$ and $-$, sometimes labelled with the potential difference it is set to.
   1. On a laboratory power supply, the positive terminal is normally red and the negative terminal is normally black.
4. An open switch is drawn as a short line hinged at one end and lifted away from the contact, showing a gap that breaks the circuit.
5. A closed switch is drawn with the line lying flat against the contact, showing that the circuit is complete.

Component symbols

1. A fixed resistor is drawn as a rectangle, and it has a set resistance that limits the current in a circuit.
2. A variable resistor is drawn as a rectangle with a diagonal arrow through it, and moving its slider changes its resistance and so changes the current.
3. A filament lamp is drawn as a circle with a cross inside it.
4. A diode is drawn as a triangle pointing at a short bar, with the wire passing straight through both.
   1. The triangle points in the direction in which conventional current is allowed to flow through the diode.
5. A light-dependent resistor (LDR) is drawn as a resistor rectangle, usually inside a circle, with two arrows pointing towards it to show light falling on it.
6. An NTC thermistor is drawn as a resistor rectangle with a diagonal line through it that ends in a short horizontal tail.
7. An ammeter is drawn as a circle containing the letter A, and a voltmeter is drawn as a circle containing the letter V.
   1. An ammeter is drawn in the line of the circuit, while a voltmeter is drawn on its own short loop that joins the circuit on either side of the component it measures.

Drawing and reading diagrams

1. Draw circuit diagrams with a ruler and a sharp pencil, using only the standard symbols.
2. Draw the wires so that each loop is closed, with no gaps except at an open switch.
3. Mark every junction with a dot, and draw the cell or battery so that its long and short lines are clear.
4. To read a diagram, start at the positive terminal and trace the path of conventional current around the circuit back to the negative terminal.
   1. Components that the current passes through one after another, with no junction between them, are in series.
   2. Components on separate branches between the same two junctions are in parallel.
5. Check which way round any diode is drawn, because a diode pointing against the conventional current from the supply blocks the current in that part of the circuit.

3.2.3 Current, potential difference and resistance

Current

1. Current, symbol $I$, is the rate of flow of charge around a circuit.
2. Current is measured in amperes, unit $A$, often shortened to amps, using an ammeter connected in series.
3. Small currents are often given in milliamperes, where $1mA=0.001A$.
4. A current can only flow when there is a closed circuit and a source of potential difference, such as a cell, to push the charge around it.

Potential difference

1. Potential difference, symbol $V$, is the energy transferred per unit charge as the charge moves between two points in a circuit.
2. Potential difference is measured in volts, unit $V$, using a voltmeter connected in parallel across the component.
3. The definition can be written as an equation, $V=\frac{E}{Q}$, where $E$ is the energy transferred in joules, $J$, and $Q$ is the charge in coulombs, $C$.
   1. This means that $1V=1J/C$, so a potential difference of $1V$ transfers $1J$ of energy for every coulomb of charge.
4. The potential difference across a cell tells you how much energy the cell gives to each coulomb of charge that passes through it.
   1. A $9V$ battery gives each coulomb of charge $9J$ of energy.
5. The potential difference across a component tells you how much energy each coulomb transfers in that component, for example to the thermal store of a resistor or away from a lamp by light.
6. The potential difference of the supply is what drives charge around a circuit, so a larger supply potential difference pushes more charge through a component every second.

Resistance

1. Resistance, symbol $R$, is a measure of how strongly a component opposes the flow of charge through it.
2. Resistance is measured in ohms, unit $\Omega$, and large resistances are often given in kilohms, where $1k\Omega =1000\Omega$.
3. In a metal, the current is a flow of free electrons, which keep colliding with the ions that make up the metal.
   1. Each collision slows the electrons down and transfers energy to the ions, so the more often the electrons collide, the greater the resistance.
4. Connecting wires are made of copper and have a very small resistance, while resistors are made to have a particular, much larger resistance.

How the three quantities are linked

1. The current through a component depends on both the potential difference across it and its resistance.
2. For a component with a fixed resistance, a larger potential difference produces a larger current, because each coulomb is pushed through with more energy.
3. For a fixed potential difference, a larger resistance produces a smaller current, because the component opposes the flow of charge more strongly.
4. The resistance of a component is found by dividing the potential difference across it by the current through it, $R=\frac{V}{I}$.
   1. To find a resistance, the potential difference and the current must be measured at the same time on the same component.
5. Current: symbol $I$, measured in amperes, $A$, with an ammeter in series.
6. Potential difference: symbol $V$, measured in volts, $V$, with a voltmeter in parallel.
7. Resistance: symbol $R$, measured in ohms, $\Omega$, and calculated from readings of $V$ and $I$.

3.2.4 Ohm's law

The resistance equation

1. Potential difference, current and resistance are linked by the equation $V=IR$, which you must be able to recall and apply.
   1. $V$ is the potential difference in volts, $V$.
   2. $I$ is the current in amperes, $A$.
   3. $R$ is the resistance in ohms, $\Omega$.
2. The equation can be rearranged to find whichever quantity is unknown.
   1. To find the current, divide the potential difference by the resistance, $I=\frac{V}{R}$.
   2. To find the resistance, divide the potential difference by the current, $R=\frac{V}{I}$.
3. Values given in milliamperes or kilohms must be changed to amperes and ohms before they are substituted.
4. For a component whose resistance does not change, the equation shows that doubling the potential difference doubles the current.
5. For a fixed potential difference, doubling the resistance halves the current.

Constant resistance

1. For some resistors, the resistance stays the same as the current through them changes.
2. A fixed resistor, or a metal wire kept at a constant temperature, behaves in this way.
3. For these components, every pair of readings gives the same value of $\frac{V}{I}$, so the current is directly proportional to the potential difference.
   1. This behaviour is described by Ohm's law, and a component that obeys it is called an ohmic conductor.
4. The condition at constant temperature is part of the law, because heating a metal changes its resistance.

Changing resistance

1. For other components, the resistance changes as the current changes.
2. The resistance of a filament lamp increases as the current through it increases.
3. The resistance of an NTC thermistor decreases as it gets warmer, and the resistance of an LDR decreases as more light falls on it.
4. For these components, the value of $\frac{V}{I}$ is different for different pairs of readings, so the current is not directly proportional to the potential difference.
5. To decide whether a component has a constant resistance, calculate $R=\frac{V}{I}$ for several pairs of readings and compare the answers.
   1. If the values agree, allowing for small measurement errors, the resistance is constant.
   2. If the values rise or fall steadily as the current increases, the resistance is changing.

Resistance of a wire

1. The resistance of a metal wire depends on its length, its thickness, the material it is made from and its temperature.
2. A longer wire has a larger resistance, because each electron makes more collisions with ions as it travels along it.
3. For a wire of one material and thickness at constant temperature, the resistance is directly proportional to the length.
   1. Doubling the length of the wire doubles its resistance.
4. A thinner wire has a larger resistance than a thicker wire of the same length and material.

3.2.5 Ohmic and non-ohmic resistors

Where resistance comes from

1. A metal is made of a regular arrangement of positive ions, with free electrons able to move between them.
2. When a potential difference is applied, the free electrons drift through the metal, forming the current.
3. As they drift, the electrons collide with the ions, which slows them down and transfers energy to the ions.
   1. The energy transferred to the ions increases the thermal store of the metal, which is why a current heats a resistor.
4. The more often the electrons collide with the ions, the greater the resistance of the metal.
5. The ions are always vibrating, and the hotter the metal, the more strongly they vibrate, so the electrons collide with them more often.

Resistors with constant resistance

1. If the temperature of a metal conductor stays the same, its ions keep vibrating by the same amount as the current changes.
2. The rate of collisions for each electron therefore does not change, so the resistance stays constant.
3. With a constant resistance, doubling the potential difference doubles the current, so the value of $\frac{V}{I}$ is the same for every pair of readings.
4. A conductor that behaves in this way is an ohmic conductor.
   1. Fixed resistors are made so that they stay close to room temperature for the currents they are designed to carry, so their resistance stays almost constant.
   2. A metal wire behaves as an ohmic conductor only while its temperature is kept constant, for example by keeping the current small.
5. If a large current heats an ordinary resistor or wire, its resistance rises, so it stops behaving as an ohmic conductor.

Resistors with changing resistance

1. In some components, the resistance changes as the current changes, so they are non-ohmic.
2. A filament lamp is the most important example.
   1. As the current through the filament increases, more energy is transferred to it every second, so its temperature rises, reaching over $2000{}^{\circ }C$ when the lamp glows brightly.
   2. The hotter filament has ions that vibrate more, so the electrons collide with them more often.
   3. More frequent collisions mean the resistance of the filament increases as the current increases.
3. Because its resistance rises, doubling the potential difference across a filament lamp less than doubles the current through it.
4. An NTC thermistor also changes its resistance as the current changes, but in the opposite way.
   1. A large current warms the thermistor, and its resistance falls as its temperature rises, so the current rises faster than the potential difference.
5. A diode has a very high resistance until the potential difference across it in the forward direction reaches about $0.6V$, after which its resistance becomes very small.
6. Non-ohmic does not mean that a component has no resistance, or that its resistance is large, but only that its resistance is not constant.

Testing whether resistance is constant

1. To test a component, measure the current through it at several different potential differences.
2. Calculate $R=\frac{V}{I}$ for each pair of readings.
3. If the values are the same, the component has a constant resistance over that range.
4. If the values increase as the current increases, as for a filament lamp, the resistance is rising.
5. If the values decrease as the current increases, as for a self-heating thermistor, the resistance is falling.

3.2.6 Designing circuits to investigate resistance

The test circuit

1. To explore how the resistance of a component behaves, the circuit must let you change the current through the component and measure both the current and the potential difference at each setting.
2. Each part of the standard test circuit has its own job.
   1. d.c. power supply or battery: provides the potential difference that drives the current.
   2. Component under test: the wire, lamp, diode, thermistor or LDR being investigated.
   3. Ammeter in series with the component: measures the current through it.
   4. Voltmeter in parallel across the component only: measures the potential difference across it, without including the ammeter or any other part.
   5. Variable resistor in series: changes the current, and so changes the potential difference across the component.
   6. Switch: lets the current be turned off between readings so that the component does not heat up.
3. Turning the variable resistor to a larger resistance gives a smaller current and a smaller potential difference across the component, while a smaller setting gives a larger current.
4. A variable power supply can be used instead of a variable resistor, by changing its output in steps.
5. To take readings for current in the opposite direction, the connections to the supply are reversed, and the readings are recorded as negative values.

Planning the investigation

1. The independent variable is usually the potential difference across the component, which you change in steps.
2. The dependent variable is the current through the component, which you measure at each step.
3. The control variables must be kept the same so that only the potential difference affects the current.
   1. For a metal wire, the control variables are its length, thickness, material and temperature.
   2. For an NTC thermistor, the temperature must be held constant at each setting, and for an LDR the light intensity must be held constant.
4. Take at least five or six readings spread evenly across the range, plus the same number in the reverse direction when the component might behave differently each way.
5. Repeat each reading and calculate a mean, ignoring any result that is clearly anomalous.

Adapting the circuit for each component

1. Wire of varying resistance: a length of resistance wire is fixed along a metre rule, and a sliding crocodile clip selects how much of it is in the circuit.
   1. The voltmeter is connected across the selected length only, and the current is kept small so the wire does not warm up.
2. Filament lamp: the potential difference is increased in steps up to the rated value printed on the lamp, and each reading is taken once the brightness has settled.
3. Diode: a protective resistor is placed in series with the diode, because a diode in the forward direction has a very small resistance and could otherwise carry a damaging current.
   1. The readings are taken in small steps around $0.6V$, where the current starts to rise sharply, and the supply is then reversed to test the other direction.
4. NTC thermistor: the thermistor is placed in a beaker of water with a thermometer, so its temperature can be changed and measured while the same measuring circuit records the current and potential difference.
5. LDR: the LDR is placed at a measured distance from a lamp in a darkened room, so the light intensity on it can be changed by moving the lamp while the circuit records the current and potential difference.

Using the readings

1. For each pair of readings, calculate the resistance using $R=\frac{V}{I}$.
2. Comparing the resistance values shows whether the resistance is constant or changes as the current changes.
3. Plotting the current against the potential difference shows the same pattern as a graph, which is the clearest way to compare components.

3.2.7 Linear and non-linear circuit components

Current-potential difference graphs

1. A current-potential difference graph shows how the current through a component changes as the potential difference across it changes.
2. The potential difference is the independent variable, so it is plotted on the horizontal axis, in volts, $V$.
3. The current is the dependent variable, so it is plotted on the vertical axis, in amperes, $A$.
4. Positive values show the current in one direction, and negative values show the current flowing the opposite way after the supply connections are reversed.
5. The resistance at any point on the graph is found by reading off $V$ and $I$ at that point and calculating $R=\frac{V}{I}$.
   1. A steeper line means more current for each volt, so it shows a smaller resistance.
   2. A shallower line means less current for each volt, so it shows a larger resistance.

Linear components

1. The graph of a linear component is a straight line through the origin.
2. Equal increases in potential difference produce equal increases in current, so the current is directly proportional to the potential difference.
3. Because $\frac{V}{I}$ is the same at every point on the line, a linear component has a constant resistance.
4. A fixed resistor and a metal wire at constant temperature are linear components.
5. The line has the same shape for negative values, because these components behave in the same way whichever direction the current flows.
6. Two linear components can be compared by the steepness of their lines, with the steeper line belonging to the component with the smaller resistance.

Non-linear components

1. The graph of a non-linear component is curved.
2. Equal increases in potential difference produce unequal changes in current, so the current is not directly proportional to the potential difference.
3. Because $\frac{V}{I}$ is different at different points on the curve, a non-linear component has a resistance that changes.
4. The direction in which the curve bends shows how the resistance changes.
   1. A curve that becomes less steep as the potential difference increases shows a resistance that is increasing.
   2. A curve that becomes steeper as the potential difference increases shows a resistance that is decreasing.
5. A filament lamp, a diode and a thermistor that warms up as the current increases are all non-linear components.

Deciding from data

1. To decide from a table of results, calculate $\frac{I}{V}$ or $\frac{V}{I}$ for each row.
   1. If the ratio is the same for every row, the component is linear.
   2. If the ratio changes steadily from row to row, the component is non-linear.
2. To decide from a graph, lay a ruler along the plotted points and check whether they lie on a straight line that passes through the origin.
3. A single point that lies well away from an otherwise smooth pattern is an anomaly and should be checked or repeated, not used to change the line.

3.2.8 Current-potential difference graphs for components

Fixed resistor and metal wire

1. The graph for a fixed resistor, or for a metal wire at constant temperature, is a straight line through the origin.
2. The line continues through the origin into the negative values, and the graph is symmetrical, because the resistor opposes the current equally in both directions.
3. The straight line shows a constant resistance, which is what a resistor is designed for, since it sets a fixed current for a given potential difference.
4. A resistor with a larger resistance gives a less steep line, because a smaller current flows for each volt.
5. If a wire carries a large current and becomes warm, the ends of its line start to bend towards the potential difference axis, because its resistance rises as it heats up.

Filament lamp

1. The graph for a filament lamp is a curve through the origin that becomes less steep as the potential difference increases in either direction, giving a shallow S-shape.
2. The graph is symmetrical, because the filament heats up in the same way whichever direction the current flows.
3. At small potential differences, the filament is cool and the graph is almost straight.
4. As the current increases, the filament gets hotter, its ions vibrate more, the electrons collide with them more often, and its resistance increases.
5. This property is tied to the job of the lamp, because the filament must become very hot to give out light, so its resistance unavoidably rises when it is working.

Diode

1. A diode only allows current to flow through it in one direction, called the forward direction.
2. In the reverse direction, the diode has a very high resistance, so the current is almost zero and the graph lies along the negative potential difference axis.
3. In the forward direction, the current stays almost zero until the potential difference reaches about $0.6V$, and then rises very steeply.
   1. Above this potential difference, the diode has a very small resistance.
4. The graph is therefore not symmetrical, which is the clearest sign of a component that treats the two directions differently.
5. Diodes are used to make sure current can only flow one way, for example to protect a device from damage if its battery is put in the wrong way round.

NTC thermistor

1. The resistance of an NTC thermistor decreases as its temperature increases.
2. A graph of resistance against temperature is a curve that falls steeply at low temperatures and then levels off at higher temperatures.
3. On a current-potential difference graph, a thermistor held at one temperature gives a straight line, and a thermistor at a higher temperature gives a steeper line because its resistance is smaller.
4. If the current is large enough to warm the thermistor, its line curves upwards and becomes steeper, because its resistance falls as it heats itself.
5. Because its resistance responds to temperature, a thermistor is used as a temperature sensor in thermostats, oven controls, fire alarms and digital thermometers.

Light-dependent resistor

1. The resistance of an LDR decreases as the light intensity falling on it increases.
2. In darkness, an LDR can have a resistance of around a million ohms, while in bright light its resistance can fall to a few hundred ohms.
3. A graph of resistance against light intensity is a curve that falls steeply at low light intensity and then levels off.
4. On a current-potential difference graph, an LDR in brighter light gives a steeper straight line, because its resistance is smaller.
5. Because its resistance responds to light, an LDR is used as a light sensor, for example in street lights that switch on at dusk and in automatic night lights.

3.2.9 Resistance in series and parallel

Net resistance

1. When several resistors are connected together, their combined effect on the circuit is described by their net resistance.
2. For a supply with a fixed potential difference, a larger net resistance means a smaller current is drawn from the supply.
3. Adding a resistor can either increase or decrease the net resistance, depending on whether it is added in series or in parallel.

Resistors in series

1. When a second resistor is added in series, there is still only one path, so all of the charge must pass through both resistors one after the other.
2. Each resistor opposes the flow of charge, so the charge meets more opposition in total than it did with one resistor.
3. The net resistance increases, and it is larger than the resistance of either resistor on its own.
4. With the same supply potential difference, the larger net resistance means the current in the circuit falls.
5. The supply potential difference is now shared between the two resistors, so the potential difference across the first resistor is smaller than before.
6. Adding a third resistor in series increases the net resistance again, so every extra resistor in series makes the current smaller.

Resistors in parallel

1. When a second resistor is added in parallel, it provides an extra path through which charge can flow.
2. Each branch is connected directly across the supply, so each resistor has the full supply potential difference across it and carries its own current.
3. The current through the first resistor does not change, but the second branch adds its own current, so the total current from the supply increases.
4. A larger total current flowing for the same potential difference means that the net resistance has decreased.
5. The net resistance of resistors in parallel is smaller than the resistance of the smallest resistor on its own.
6. Every extra resistor added in parallel adds another path, so the net resistance keeps falling and the total current keeps rising.

Lamps in series and in parallel

1. Identical lamps connected in series to a supply glow more dimly than one lamp on its own.
   1. The net resistance is larger, so the current is smaller, and the supply potential difference is shared, so each lamp transfers energy more slowly.
2. Identical lamps connected in parallel to the same supply each glow as brightly as one lamp on its own.
   1. Each lamp has the full supply potential difference across it and carries the same current as a single lamp would.
3. The lamps in parallel draw a larger total current, so a battery powering them goes flat more quickly than the same battery powering lamps in series.

3.2.10 Series and parallel circuit problems

Calculating in series circuits

1. In a series circuit, the current is the same through every component, so one current value is used for every component.
2. The total resistance of components in series is the sum of their resistances, $R_{total}=R_1+R_2$.
3. The supply potential difference is shared, so the potential differences across the components add up to it, $V_{supply}=V_1+V_2$.
4. The equation $V=IR$ can be applied to the whole circuit or to one component, as long as all three values belong to the same part.
   1. For the whole circuit, use the supply potential difference, the circuit current and the total resistance.
   2. For one component, use the potential difference across that component, the current through it and its own resistance.
5. The component with the larger resistance has the larger potential difference across it, because the same current flows through both and $V=IR$.

Calculating in parallel circuits

1. In a parallel circuit, the potential difference across each branch is the same as the supply potential difference.
2. The current in each branch is found separately, using $I=\frac{V}{R}$ with the supply potential difference and the resistance of that branch.
3. The total current from the supply is the sum of the branch currents, $I_{total}=I_1+I_2$.
4. The total resistance of a parallel circuit is found by dividing the supply potential difference by the total current, $R_{total}=\frac{V_{supply}}{I_{total}}$.
   1. This total resistance is always smaller than the resistance of any single branch.
5. The branch with the smaller resistance carries the larger current, because each branch has the same potential difference across it.

Solving circuit problems

1. Decide first whether the components are in series, in parallel, or a mixture of both.
2. Write down the values you know and label each one with the component it belongs to.
3. Use the series and parallel rules to find any potential difference or current that is shared or added.
4. Use $V=IR$ for a single component or for the whole circuit to find the remaining unknown.
5. Check the answer, for example by confirming that the series potential differences add up to the supply or that the branch currents add up to the total current.

Circuits with sensors

1. The same rules apply when a circuit contains an NTC thermistor or an LDR, except that the resistance of the sensor depends on its surroundings.
2. The resistance of a thermistor at a given temperature, or of an LDR at a given light intensity, is usually read from a graph and then used in the calculation.
3. When a thermistor is in series with a fixed resistor, a rise in temperature lowers the thermistor resistance, so the circuit current increases and the potential difference across the thermistor falls.
4. A photocell acts as a supply rather than a resistor, and the power it delivers is found by multiplying the potential difference across it by the current it drives, $P=VI$.

3.2.11 Circuits for measurement and testing

Meters for measuring circuits

1. Measuring circuits are built so that the meters record the current and potential difference without changing what they measure.
2. An ammeter is placed in series and has a very low resistance, so adding it to a circuit barely changes the current.
3. A voltmeter is placed in parallel and has a very high resistance, so it draws almost no current from the component it measures.
4. A multimeter combines several meters in one, and a dial selects whether it measures current, potential difference or resistance.
   1. The black lead goes into the socket marked COM, while the red lead goes into the socket for volts and ohms or the socket for amperes, depending on the measurement.
   2. On the resistance setting, the multimeter passes a small current from its own battery, so the component must be disconnected from the circuit first.
5. Each setting has a range, such as $20V$ or $200mA$, and the best range is the smallest one that is still larger than the value being measured.

Accurate readings

1. The resolution of a digital meter is set by its last digit, so a meter reading $1.23V$ has a resolution of $0.01V$.
   1. Choosing a smaller range usually gives more digits after the decimal point, and so a better resolution.
2. Before a meter is used, check for a zero error by reading it with no current or no potential difference applied.
   1. A zero error must be subtracted from every reading, because it shifts all the readings by the same amount.
3. On an analogue meter, read the needle with your eye directly in line with it, so that parallax does not make the reading too high or too low.
4. Readings change if the component warms up, so close the switch only while taking each reading.
5. Take each reading more than once and calculate a mean, ignoring any value that is clearly anomalous.

Testing for faults

1. An open circuit has a break in it, such as a broken wire, a loose connection or a blown filament, so there is no current anywhere in that loop.
2. A short circuit is a low-resistance path that lets the current bypass a component, so that component stops working and the current in the rest of the circuit rises.
3. A continuity tester, made from a cell and a buzzer or lamp, or a multimeter on its resistance setting, shows whether a wire, fuse or switch provides a complete path.
   1. A reading of nearly $0\Omega$, or a sounding buzzer, shows a complete path, while a very large reading shows a break.
4. A voltmeter can find the faulty part of a series circuit without disconnecting anything.
   1. Across a working component in a circuit with a break, the voltmeter reads zero, because no current flows through it.
   2. Across the broken component, the voltmeter reads the full supply potential difference, because the voltmeter completes the circuit around the gap.
5. A component that has been short-circuited has a potential difference of zero across it, even though the supply is working.

Circuits that measure

1. A sensor in series with a fixed resistor turns a change in the surroundings into a change in potential difference, which a voltmeter can read.
2. In a temperature-sensing circuit, an NTC thermistor is connected in series with a fixed resistor and a d.c. supply, with a voltmeter across the fixed resistor.
   1. As the temperature rises, the resistance of the thermistor falls, so the current increases.
   2. The fixed resistor now takes a larger share of the supply potential difference, so the voltmeter reading rises.
3. A light-sensing circuit works in the same way, with an LDR in place of the thermistor, so the voltmeter reading rises as the light gets brighter.
4. The circuit is calibrated by recording the voltmeter reading at several known temperatures or light intensities, so that any later reading can be converted back into a temperature or light intensity.

3.2.12 Power transfer in circuit devices

Energy transfer in a circuit

1. A circuit is a way of transferring energy from the supply to the components, with the current acting as the pathway.
2. The supply transfers energy to the charge, and each component transfers that energy electrically to another store or away from the circuit.
   1. In a resistor or a heater, the energy is transferred to the thermal store of the component and then by heating to the surroundings.
   2. In a filament lamp, the energy is transferred to the thermal store of the filament and away by light and by heating.
   3. In a motor, most of the energy is transferred to the kinetic store of the spinning parts.
3. The energy transferred when charge $Q$ passes through a potential difference $V$ is given by $E=QV$.
   1. $E$ is the energy transferred in joules, $J$, $Q$ is the charge in coulombs, $C$, and $V$ is the potential difference in volts, $V$.
4. The current is not used up in a component, because it is the energy carried by the charge that is transferred.

Power

1. Power tells you how quickly a device transfers energy, not how much energy it transfers in total.
2. Power is the energy transferred divided by the time taken, $P=\frac{E}{t}$, which you can rearrange to $E=Pt$.
   1. $P$ is the power in watts, $W$, $E$ is the energy in joules, $J$, and $t$ is the time in seconds, $s$.
3. One watt is one joule per second, so $1W=1J/s$, and a $60W$ lamp transfers $60J$ of energy every second.
4. Large powers are given in kilowatts, where $1kW=1000W$, so a $2kW$ kettle transfers $2000J$ every second.

Power, potential difference and current

1. The power of any circuit device depends on the potential difference across it and the current through it.
   1. The potential difference is the energy transferred by each coulomb of charge.
   2. The current is the number of coulombs passing each second.
   3. Multiplying energy per coulomb by coulombs per second gives energy per second, which is the power.
2. This gives the equation $P=VI$, where $P$ is in watts, $V$ is in volts and $I$ is in amperes.
3. Doubling either the potential difference or the current, with the other kept the same, doubles the rate at which the device transfers energy.
4. Substituting $V=IR$ into $P=VI$ gives a second equation, $P=I^{2}R$, which is useful when the current and the resistance are known.
   1. Because the current is squared, doubling the current through a fixed resistor makes it transfer energy four times as quickly, which is why large currents heat wires so strongly.

Energy over time

1. The total energy transferred by a device depends on its power and on how long it is used, $E=Pt$.
2. With power in watts and time in seconds, the energy comes out in joules.
3. With power in kilowatts and time in hours, the energy comes out in kilowatt-hours, $kW\; h$, which is the unit used on electricity bills.
   1. One kilowatt-hour is equal to $1000W\times 3600s=3600000J$.
4. A device with a higher power transfers energy faster, but a lower-power device left on for longer can still transfer more energy in total.
5. The power rating on an appliance, such as $230V$, $2.3kW$, gives the power it transfers when connected to the stated potential difference.

3.2.13 Energy and power in resistor circuits

Equivalent resistance

1. Several resistors in series can be replaced, in a calculation, by one resistor with the same overall effect on the circuit.
2. For resistors in series, the equivalent resistance is the sum of the individual resistances, $R_{eq}=R_1+R_2+R_3$.
3. This works because the same current flows through every resistor, and the potential differences across them add up to the supply potential difference.
4. The current drawn from the supply is found by dividing the supply potential difference by the equivalent resistance, $I=\frac{V_{supply}}{R_{eq}}$.
5. Adding another resistor in series increases the equivalent resistance, so for the same supply the current falls.

The equations you need

1. All of these equations must be recalled, so learn each one with its units.
2. Charge: $Q=It$, with charge in coulombs, $C$, current in amperes, $A$, and time in seconds, $s$.
3. Potential difference: $V=IR$, with potential difference in volts, $V$, and resistance in ohms, $\Omega$.
4. Energy from charge: $E=QV$, with energy in joules, $J$.
5. Power from potential difference: $P=VI$, with power in watts, $W$.
6. Power from resistance: $P=I^{2}R$.
7. Energy from power: $E=Pt$, with energy in joules and time in seconds, or energy in $kW\; h$ with power in $kW$ and time in hours.

Solving series circuit problems

1. Add the resistances to find the equivalent resistance of the circuit.
2. Use $I=\frac{V}{R_{eq}}$ with the supply potential difference to find the current, which is the same through every resistor.
3. Use $V=IR$ with each resistor's own resistance to find the potential difference across it.
4. Use $P=VI$ or $P=I^{2}R$ to find the power of each resistor or of the whole circuit.
5. Use $E=Pt$ or $E=QV$ to find the energy transferred in a given time, and $Q=It$ to find the charge that flows.
6. Check the answers using conservation of energy.
   1. The potential differences across the resistors add up to the supply potential difference.
   2. The powers of the resistors add up to the power supplied by the source.

Comparing resistors in series

1. Every resistor in a series circuit carries the same current, so by $P=I^{2}R$ the resistor with the larger resistance transfers energy at the greater rate.
2. The larger resistor also has the larger potential difference across it, so each coulomb transfers more energy in that resistor.
3. When an extra resistor is added in series, the current falls, so the power of each of the original resistors decreases.