Series and parallel circuits

This question is about the batteries used to store energy in electric vehicles. Each battery is a series of modules. Each module is an array of identical cells.

Table 2 shows the properties of one of these cells.

Calculate the terminal potential difference of the cell at its maximum current.

Figure 6 shows an array of 20 of these cells connected to form a module.

Calculate the internal resistance of the module shown in Figure 6.

Figure 7 shows the battery for an electric vehicle. This battery has an emf of 352 V and gives a maximum current of 500 A.

The battery consists of a series of identical modules. These modules are different from the module shown in Figure 6.

The individual cells used in each module have the properties shown in Table 2.

Deduce the number of cells in each module.

Two thin metal rods W₁ and W₂, each of length L and diameter d, are connected in series.

A single metal rod X, also of length L and diameter d, has the same resistance as the

What is the resistivity of X ?

In a circuit, the resistance of resistor R₁ is double the resistance of resistor R₂. The internal resistance of the battery is negligible.

When the switch is open, the voltmeter reads 12 V and the ammeter reads 12 mA.

What are the readings on the voltmeter and ammeter when the switch is closed?

Figure 3 shows a circuit that causes a loudspeaker to emit ‘clicks’ at a constant rate. The time T between successive clicks depends on the resistances R₁ and R₂ and the capacitance C. You do not need to know any details about the electronic system.

A student uses a stopwatch to determine T. Table 1 shows the student’s repeated measurements of 20 T.

Suggest why the student measures 20 T.

Suggest why the student repeats their measurements of 20 T.

Determine the percentage uncertainty in the student’s result for T. Assume that all the data in Table 1 are valid.

The student determines T for different values of R₂. The values of R₁ and C are not changed.

The student produces the graph in Figure 4.

It can be shown that

The student has access to a range of fixed and variable resistors. The student wants to modify the circuit in Figure 3 so that T can be varied continuously from a minimum of 0.70 s to a maximum of 1.25 s.

Show how the student can achieve this by combining two resistors to make R₂. You should use information from Figure 4.

Figure 5 shows apparatus to investigate how the resistance R of a lamp filament varies with the absolute temperature T of the filament.

The lamp is inside an oven.

The oven temperature is increased to a steady value. The filament is at the same temperature as the inside of the oven.

The temperature T in kelvin is determined using the thermometer. An ohm-meter is connected to the filament to determine R at temperature T.

The oven temperature is increased to new steady values and further readings of T and R are recorded.

Figure 6 shows the variation of R with T.

It can be shown that:

where:

R₀ is the resistance of the filament at temperature T₀ α is a constant.

Determine α using the graph in Figure 6. State an appropriate SI unit for your answer.

In a different investigation, a student investigates how the electrical power P transferred in the same filament lamp varies for values of T greater than 800 K.

The student assumes that α is constant for all values of T.

Figure 7 shows the circuit that the student uses. The temperature of the filament is controlled by the energy supplied to it from the battery.

Complete Table 2.

It can be shown that P₁ Tⁿ where n is an integer.

Determine n.

In your answer you should:

• use Table 3 to organise the data you will plot

• plot a suitable graph on Figure 9.

The circuit in Figure 5 is used as part of a temperature sensor. The battery has an emf of 6.5 V and negligible internal resistance.

The initial temperature of the thermistor is 22 °C. At this temperature the resistance of the thermistor is 350 Ω and the circuit current is 12 mA.

Calculate the resistance of the variable resistor.

The resistance R of the thermistor at temperature θ in K is given by:

where R₀ is the resistance at the initial temperature θ₀ in K, and B is a constant.

The temperature of the thermistor is increased to 318 K. The variable resistor is adjusted so that the circuit current is again 12 mA. The potential difference across the thermistor is now 3.2 V.

Determine B. State an appropriate unit for your answer.

Explain why the current in the thermistor needs to be controlled.

Explain how ammeters and voltmeters can be used in the circuit in Figure 5 to demonstrate the conservation of charge and the conservation of energy.

Refer to points X, Y and Z in your answer.

In Figure 2 the cell has emf ε and internal resistance r.

The current in the circuit is I.

The potential difference (pd) across R₁ is V₁ and the pd across R₂ is V₂.

Explain how the law of conservation of energy applies in this circuit. You should consider the movement of one coulomb of charge around the circuit.

Figure 3 shows a variable resistor made with a thin conducting layer on an insulating base.

The conducting layer has constant width and thickness and has connections at the ends A and B. C is a sliding contact that can move along the surface of the conducting layer between A and B.

Figure 4 shows a circuit that uses the variable resistor as a potential divider.

The variable resistor is connected to a battery of emf 3.00 V and internal resistance r The resistance of the conducting layer between A and B is 125 Ω.

The sliding contact C is moved to end B of the variable resistor. The switch is closed. The digital voltmeter reads 2.89 V.

Show that r is approximately 4.8 Ω.

C is set at of the distance between A and B. The thickness of the conducting layer is uniform so the resistance between A and C is 25.0 Ω.

Determine the voltmeter reading at this setting.

Figure 5 shows a variable resistor similar to the one shown in Figure 3 but with the following three manufacturing faults:

• at P the conducting layer changes in thickness so that AP is thinner than PB

• at Q there is a scratch into the surface of the conducting layer and across its full width

• from R to B the conducting connector is laid over the conducting layer.

The width of the conducting layer is constant.

A pd of 3.0 V is applied across A and B. C is moved from A to B.

Sketch, on the axes in Figure 6, a graph to show how the pd between A and C varies as C is moved from A to B.

A box with four terminals is connected to a cell and two ammeters. The top left terminal

Each of the boxes A to D is connected into the circuit in turn. All the resistors have equal resistance.

Which box gives the same reading on both ammeters?

A resistor dissipates 100 W when connected across a 25 V supply with negligible internal resistance. The supply output is reduced to 20 V and the resistor is replaced so that the power dissipated is still 100 W.

What is the percentage decrease in resistance?

Figure 4 is a plot of current – voltage data for a filament lamp L.

Figure 4

The current I was measured as the voltage V across L was increased at a steady rate. These data were obtained using a current sensor and a voltage sensor connected to a data logger. The logger recorded data at a rate of 2.5 Hz.

Determine, in V s⁻¹, the rate of increase of V.

State two advantages of using data logging for this experiment.

Figure 5 shows two circuits that can be used to collect current–voltage data.

The dc supply has an emf of 12 V and negligible internal resistance. The current sensor and the voltage sensor behave as ideal meters.

In circuit 1:

• X is used as a variable resistor with a maximum resistance of 14.9 Ω

• when X is set to maximum resistance, the resistance of L is 2.3 Ω.

In circuit 2, X is used as a potential divider.

Discuss, with reference to circuit 1 and circuit 2, whether either circuit can produce all the data shown in Figure 4. Support your answer with a calculation.

Table 2 shows some values of V that are plotted on Figure 4 and corresponding results for I and for the power P dissipated in L.

Complete Table 2.

Plot on Figure 6 a graph of P against V. You should use only the data in your completed Table 2.

L is connected to a 12 V power supply of negligible internal resistance. L then dissipates its rated power P_r.

A second lamp, identical to L, is now connected in series with L.

Determine the percentage of P_r that is dissipated in this circuit.

A student assembles the circuit in Figure 6.

The battery has an internal resistance of 2.5 Ω.

Show that the resistance of the 6.2 V, 4.5 W lamp at its working potential difference (pd) is about 9 Ω.

The terminal pd across the battery is 6.2 V.

Calculate the emf of the battery.

The student makes a variable resistor to control the brightness of the lamp.

Figure 7 shows her circuit.

She uses a resistance wire with a diameter of 0.19 mm to make the variable resistor. A 5.0 m length of this wire has a resistance of 9.0 Ω.

Calculate the resistivity of the wire.

Figure 8 shows the 5.0 m length of wire wrapped around a tube to make the variable resistor.

Two plugs connect the variable resistor into the circuit. A moveable copper contact is used to vary the length of wire in series with the lamp. When the contact is placed on the tube at one particular position, the lamp is dim. The contact is then moved slowly to the right as shown in Figure 8.

Explain, without calculation, what happens to the brightness of the lamp as the contact is moved.

The student now makes a different circuit by connecting the variable resistor in parallel with the lamp.

The contact is returned to its original position on the tube as shown in Figure 8 and the lamp is dim. The contact is again slowly moved to the right.

Explain, without calculation, what happens to the brightness of the lamp as the contact is moved.

Which value of resistance cannot be made by combining three 10 Ω resistors?

Figure 11 shows how the resistance of an LDR varies with light intensity.

The LDR is used as part of an alarm system in a dim room. Figure 12 shows one proposal for a sensor circuit for this system.

The power supply to the sensor has an emf of 5.0 V and a negligible internal resistance. A negligible current is drawn from the sensor circuit by the alarm subsystem.

A light beam illuminates the LDR. When the light beam is broken the LDR is not illuminated by the light beam. This causes the alarm to sound.

Table 3 shows how the light intensity at the LDR changes.

Show that the current in the sensor circuit when the LDR is not illuminated by the light beam is approximately 16 µA.

The alarm sounds when the potential difference V s across the LDR changes by more than 25% of the power supply emf.

Discuss whether the circuit shown in Figure 12 is suitable. Support your answer with a calculation.

Two identical batteries each of emf 1.5 V and internal resistance 1.6 Ω are connected in parallel. A 2.4 Ω resistor is connected in parallel with this combination.

What is the current in the 2.4 Ω resistor?

Two resistors X and Y are connected in series with a power supply of emf 30 V and negligible internal resistance. The resistors are made from wire of the same material. The wires have the same length. X uses wire of diameter d and Y uses wire of diameter 2 d.

What is the reading on the voltmeter?

A light-emitting diode (LED) emits light over a narrow range of wavelengths.

Two LEDs L_G and L_R are adjusted to give the same maximum light intensity. L_G emits green light and L_R emits red light.

Figure 3 shows how the light output of the LEDs varies with the wavelength λ.

Figure 3

Light from L_R is incident normally on a plane diffraction grating.

Determine N, the number of lines per metre on the grating.

Suggest one possible disadvantage of using the fifth-order maximum to determine N

Figure 4 shows part of the current−voltage characteristics for L_R and L_G.

When the linear part of the characteristic is extrapolated, the point at which it meets the horizontal axis gives the activation voltage V_A for the LED.

Determine, using Figure 4, V_A for L_R.

It can be shown that:

where h = the Planck constant.

Deduce a value for the Planck constant based on the data given about the LEDs.

Figure 5 shows a circuit with L_R connected to a resistor of resistance R.

The power supply has emf 6.10 V and negligible internal resistance. The current in L_R must not exceed 21.0 mA.

Deduce the minimum value of R.

Figure 9 shows some of the apparatus used in a demonstration of electrical power transmission using a dc power supply.

A power supply of emf 12 V and negligible internal resistance is connected to three identical 12 V, 1.5 W lamps in parallel.

Show that the resistance of one of the lamps when it is operating at 12 V is about 100 Ω.

Initially the power supply is connected to the lamps using two short copper wires of negligible resistance.

Calculate the current in the power supply.

The two short copper wires are replaced with two long constantan wires.

Show that the resistance of each length of constantan wire is about 50 Ω.

length of each constantan wire = 2.8 m diameter of constantan wires = 0.19 mm resistivity of constantan = 4.9 × 10⁻⁷ Ω m

The demonstration is intended to show that the lamps are significantly dimmer when connected using the long constantan wires than when using the short copper wires.

Discuss whether the demonstration achieves this. Support your answer with suitable calculations.

Scientists and engineers are investigating the use of superconductors in electrical transmission.

Discuss one advantage and one difficulty when using superconductors in electrical transmission over long distances.

An electric motor lifts a load of weight W through a vertical height h in time t. The potential difference across the motor is V and the current in it is I.

What is the efficiency of the motor?

A resistor of resistance R and three identical cells of emf E and internal resistance r are connected as shown.

What is the current in the resistor?

Figure 10 shows a partly-completed circuit used to investigate the emf ε and the internal resistance r of a power supply.

The resistance of P and the maximum resistance of Q are unknown.

Figure 10

Complete Figure 10 to show a circuit including a voltmeter and an ammeter that is suitable for the investigation.

Describe

• a procedure to obtain valid experimental data using your circuit

• how these data are processed to obtain ε and r by a graphical method.

Figure 11 shows a different experiment carried out to confirm the results for ε and r.

Initially the power supply is connected in series with an ammeter and a 22 Ω resistor. The current I in the circuit is measured.

The number n of 22 Ω resistors in the circuit is increased as shown in Figure 11. The current I is measured after each resistor is added.

It can be shown that

Figure 12 on page 22 shows a graph of the experimental data.

Figure 12

Show that ε is about 1.6 V.

Figure 13 shows the circuit when four resistors are connected.

Show, using Figure 12, that the current in the power supply is about 0.25 A.

Deduce, for the circuit shown in Figure 13,

• the potential difference (pd) across the power supply

Figure 14 shows the plots for n = 1 and n = 14

Three additional data sets for values of n between n = 1 and n = 14 are needed to complete the graph in Figure 14.

Suggest which additional values of n should be used. Justify your answer.

The experiment is repeated using a set of resistors of resistance 27 Ω.

The relationship between n and I is now

Show on Figure 14 the effect on the plots for n = 1 and n = 14 You do not need to do a calculation.

Figure 9 shows a practical circuit in which a variable resistor is used to control the brightness of a lamp. The voltmeter reading is monitored as the variable resistor is adjusted to make the lamp brighter.

Explain why the reading on the voltmeter decreases as the brightness of the lamp increases.

The variable resistor is adjusted so that the lamp is at its brightest. The reading V₁ on the voltmeter is noted. A second identical cell is then connected in parallel with the cell in Figure 9. The new reading V₂ on the voltmeter is noted.

Explain why V₂ is greater than V₁.

The diagram shows part of a circuit and the currents in the circuit.

What is the potential difference between point P and earth?

A voltmeter has a resistance of 4.0 kΩ and reads 1.0 V for every scale division on the meter.

A power supply of emf 20 V and negligible internal resistance is connected across this voltmeter and a resistor in series. The voltmeter reads two divisions.

What is the value of the resistor?

Two cylindrical wires P and Q are of equal length and made of the same material. The diameter of P is greater than that of Q.

P and Q are connected in series and the ends of this arrangement are connected to a power supply.

Which two quantities are the same for P and Q ?

An electric motor lifts a load of weight W through a vertical height h in time t. The potential difference across the motor is V and the current through it is I.

What is the efficiency of the motor?

Figure 7 shows a circuit designed by a student to monitor temperature changes.

The supply has negligible internal resistance and the thermistor has a resistance of 750 Ω at room temperature. The student wants the output potential difference (pd) at room temperature to be 5.0 V

The 0.25 kΩ resistor is made of 50 turns of wire that is wound around a non-conducting cylinder of diameter 8.0 mm

Determine the area of cross-section of the wire that has been used for the resistor.

The student selects a resistor rated at 0.36 W for the 0.25 kΩ resistor in Figure 7.

Determine whether this resistor is suitable.

Determine the value of R that the student should select.

Give your answer to an appropriate number of significant figures.

State and explain the effect on the output pd of increasing the temperature of the thermistor.

A mobile phone operates at a constant power of 200 mW It has a 3.7 V lithium-ion battery that has a charge capacity of 9400 C

What is the time taken for the battery to discharge completely?

The two resistors shown are both uniform cylinders of equal length made from the same conducting putty.

The diameter of Y is twice that of X. The resistance of Y is R.

What is the total resistance of the combination?

The engineer first investigates the emf and internal resistance of a solar cell under typical operating conditions.

The engineer determines how the potential difference across the solar cell varies with current. The results are shown in the graph in Figure 4.

The engineer uses the graph to deduce that when operating in typical conditions a single solar cell produces an emf of 0.70 V and has an internal resistance of

Explain how the engineer uses the graph to obtain the values for the emf and internal resistance of the solar cell.

To operate effectively the lamp in the road sign needs a minimum current of 75 mA. At this current the resistance of the filament lamp is 6.0 .

The engineer proposes to try the two circuits shown in Figure 5 and Figure 6.

Deduce, using calculations, whether the circuits in Figure 5 and Figure 6 are suitable for this application.

Solar cells convert solar energy to useful electrical energy in the road sign with an efficiency of 4.0%. The solar-cell supply used by the engineer has a total surface area of 32 cm².

Calculate the minimum intensity, in W m^–2, of the sunlight needed to provide the minimum current of 75 mA to the road sign when it has a resistance of 6.0 .

The current in the circuit is 20 mA.

What is the value of the resistor R ?

Which statement correctly describes the variation of R n as n increases?

The resistance of wire P is R.

What is the total resistance of the wires when they are connected in parallel?