EMF and internal resistance

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.

A 12 Ω resistor is connected across the terminals of a battery of emf 2.0 V and internal resistance 4.0 Ω.

What is the pd across the resistor?

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 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.

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?

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?

In the circuit, the reading of the voltmeter is V. When the switch is closed the reading becomes .

What is the internal resistance of the cell?

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 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 .