Electromotive Force & Internal Resistance (AQA A Level Physics)

Figure 1 below shows two circuits X and Y that were used by a student to test a battery of four identical cells.

In circuit X there was no load resistor and in circuit Y a load resistor was connected. You can assume that the meters in the circuits are ideal.

Figure 1

Explain why there is a difference in voltages recorded in the two circuits.

Calculate the internal resistance of a single cell.

One of the cells in the battery is reversed.     

Determine the new reading on the ammeter in circuit Y.

In circuit Y, the resistance of the load resistor R is altered so that a series of values on the voltmeter and the corresponding values of the current on the ammeter are obtained. 

Figure 2

Using the axes in Figure 2, sketch the graph you would expect to obtain as R is changed. State how the values of ε and r can be obtained from the graph.

A battery of e.m.f. 30.0 V and internal resistance, r, is connected in the circuit shown in Figure 1. 

Figure 1

The current in the battery is 1.5 A. 

Calculate the internal resistance r.

Calculate the energy dissipated in the battery in 3.0 minutes.

Calculate the percentage of the total energy transformed by the battery that is dissipated in the battery in 3.0 minutes.

The internal resistance of the battery affects the efficiency of the transfer of energy from the battery to the circuit. 

Explain what causes internal resistance and why this affects the efficiency.

A cell of e.m.f., ∈, and internal resistance, r, is connected to a variable resistor R. The current through the cell and the terminal p.d. of the cell are measured as R is decreased. The circuit is shown in Figure 1 below.

Figure 1

The graph in Figure 2 below shows the results from the experiment. 

Figure 2

State the relationship between the terminal pd and current and explain why this relationship occurs.

Use the graph in Figure 2 to find the e.m.f., ∈, and the internal resistance, r, of the cell.

Draw a line on the graph in Figure 2 that shows the results obtained from a cell with the same e.m.f. but half the internal resistance of the first cell. Label your graph A.

Draw a line on the graph in Figure 2 that shows the results obtained from a cell with the same e.m.f. but negligible internal resistance. Label your graph B.

A battery is connected to an 8.0 Ω resistor as shown in Figure 1 below. The e.m.f. of the battery is 15 V. 

Figure 1

When the switch is open the voltmeter reads 15 V and when it is closed it reads 14.3 V. 

Explain why the readings are different.

Calculate the internal resistance of the battery.

The circuit diagram in Figure 2 shows that the 8.0 Ω resistor is now connected in parallel with an unknown resistor, R. The battery now supplies a current of 2.0 A and has the same internal resistance r as the circuit in Figure 1. 

Figure 2

Calculate the p.d. across the 8.0 Ω resistor.

Calculate the resistance of R. 

A very high resistance voltmeter reads 10.5 V when it is connected across the terminals of a power supply. 

Explain why the reading on the voltmeter is equal to the emf of the power supply.

A battery of e.m.f. 10.5 V and internal resistance r is connected in a circuit as shown in Figure 1 with three identical 15 Ω resistors. A current of 0.30 A flows through the battery. 

Figure 1

Calculate the potential difference between points A and B in the circuit.

Calculate the internal resistance of the battery.

Calculate the fraction of the energy supplied by the battery that is dissipated within the battery in 5.0 s.

A uniform wire of length 80 cm and radius 0.50 mm is connected in series with a cell of e.m.f. 3.0 V and an internal resistance of 0.70 Ω as shown in Figure 1. 

Figure 1

The resistivity of the metal used to make the wire is 1.1 × 10^–6 Ω m. 

Determine the current that flows in the cell.

A voltmeter is connected at X, with a movable probe C, such that the voltmeter is able to read the potential difference across the wire at different points between X and Y, as shown in Figure 2a.  

Figure 2a

Sketch a graph on the set of axes provided in Figure 2b which shows how the potential difference V varies between X and Y as the sliding contact C moves from X to Y. 

Figure 2b

Your sketch should include all appropriate labels.

The voltmeter in Figure 2a is replaced with a cell of e.m.f. 1.5 V and internal resistance 0.50 Ω and an ammeter as shown in Figure 3. 

Figure 3

The moveable contact can again be connected to any point along the wire XY. 

At point D, there is zero current in the ammeter. 

Calculate the length of XD.

Figure 1 shows a circuit which can be used to find investigate the internal resistance r of a power supply. In this case, a battery consisting of six dry cells in series, each of e.m.f. = 0.5 V, is connected to an oscilloscope:   

Figure 1a

Figure 1b represents the trace shown on the oscilloscope screen when both of the switches  and are open:

Figure 1b

The vertical sensitivity of the oscilloscope is set at 1.5 V div^–1. 

With reference to Figure 1b, discuss what happens to the trace shown on the oscilloscope screen when switch  is closed.  

Use Figure 1b to draw the trace on the oscilloscope screen when both switches  and are closed. 

Explain your answer.

Calculate the internal resistance of the battery if the vertical distance between the traces in part (b) and part (c) is measured to be half a division. 

Determine the current in the cell that would move the trace shown on the oscilloscope screen back to its original position shown in Figure 1b. 

Assume both switches  and  remain closed.

The Maximum Power Transfer theorem says the maximum amount of electrical power is dissipated in a load resistance  when it is exactly equal to the internal resistance of the power source r. 

Figure 1a shows a circuit used to investigate maximum power transfer. A variable resistor, which acts as the load resistance , is connected to a power source of e.m.f.  and internal resistance r, along with a switch S and an ammeter and voltmeter: 

Figure 1a

Figure 1b shows the graph obtained for the power P dissipated in R_L as the potential difference V across  is varied: 

Figure 1b

Assuming the Maximum Power Theorem is valid, use Figure 1b to determine the internal resistance of the power source.

Hence, show that the e.m.f. of the power supply is 9 V.

Identify what happens to each of the following quantities as the value of the load resistance becomes infinitely large:         

i)       Current

ii)      Potential difference across 

iii)     Power dissipated in 

It can be shown that the power P dissipated in the load resistance  is zero when the load resistance is zero.

Using this, together with the result of the Maximum Power Theorem given in part (a) and your answer to part (c)(iii), sketch a graph on the axes provided in Figure 2 to show how the power dissipated P varies with load resistance .

Label the position of the internal resistance, r. 

Figure 2

Understanding the properties of e.m.f. and internal resistance can help the design decisions of architects and electrical engineers. 

In an experiment to investigate power dissipation across two lamps L₁ and L₂, an engineer connects them in a series circuit to a cell of e.m.f. 45 V and internal resistance 7 Ω. This is shown in Figure 1: 

Figure 1

The lamps L₁ and L₂, have a resistance of 10 Ω and 25 Ω respectively. 

Calculate the percentage difference between the power generated by the cell and the power dissipated in the two lamps L₁ and L₂. 

Suggest a reason for this percentage difference.

The engineer wishes to maximise the power dissipated across each lamp and explores various alternatives to the circuit shown in Figure 1.  

Suggest and explain, using appropriate calculations, how the engineer should arrange the lamps L₁ and L₂ such that the power dissipated in each lamp is maximised.

The engineer comes up with a theoretical problem, which involves arranging a large number of identical lamps in parallel with each other, as illustrated in Figure 2 below. 

Figure 2

The lamps in Figure 2 are connected to a cell of e.m.f.  and internal resistance r. 

Discuss the effect on the terminal pd supplied by the cell, and hence on the lamps in Figure 2, as more lamps are added in parallel.