Resistivity

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 ?

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.

This question is based on a method to determine the resistivity of a wire (required practical activity 5).

Figure 1 shows a micrometer screw gauge.

Figure 2 shows an enlarged view of the scales.

State, in mm, the resolution of the main scale.

What is the reading on the micrometer?

Tick () one box.

A wire X is placed in the gap between the anvil and the spindle.

State and explain how this gap is closed just before taking a reading of the diameter of X.

Figure 3 shows a circuit used to determine the resistance per metre of wire X.

Two terminals are used to mount X on a ruler. Clips are used to connect a voltmeter across the 1.2 Ω resistor. When the switch is closed, the voltmeter reading is 931 mV.

The switch is then opened and the voltmeter is connected to X as shown in Figure 4.

When the switch is closed, the voltmeter reading is 397 mV.

Show that, for the arrangement in Figure 4, the resistance R of the wire between the clips is about 0.5 Ω.

The length of wire between the clips is L. Values of R are determined for different values of L. Figure 5 shows these data.

Determine the resistance per metre of X.

Table 1 shows the resistance per metre of various metal wires. The diameter of X is one of the values of d shown in Table 1.

Identify the metal used for X. Go on to determine the resistivity of the metal. State an appropriate SI unit for your answer.

A student adds error bars for R and L to each point on Figure 5.

She estimates that

• each value of R has a percentage uncertainty of 6%

• each value of L has an absolute uncertainty of 5 mm.

Compare her error bars for the point at L = 209 mm with her error bars for the point at L = 388 mm.

Outline how error bars are used to determine the uncertainty in the gradient of a linear graph.

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.

Which statement about a superconducting metal is correct?

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.

Conductive putty can easily be formed into different shapes to investigate the effect of shape on electrical resistance.

A student uses vernier callipers to measure the diameter d of a uniform cylinder made of the putty.

Suggest one problem with using callipers to make this measurement.

Table 1 shows the calliper measurements made by a student.

Show that the percentage uncertainty in d is about 2.4%. Assume that all the data are valid.

The length of the cylinder is 71 ± 2 mm.

Determine the uncertainty, in mm³, in the volume of the cylinder.

A student is given some putty to form into cylinders.

To find the resistance of a cylinder, metal discs are placed in contact with the ends of the cylinder and connected to a resistance meter.

Figure 7 shows the apparatus.

The student forms the putty into cylinders of different lengths, each of volume

The student plots the graph shown in Figure 8.

Determine ρ. State an appropriate SI unit for your answer.

Figure 8

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?

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.

In the circuit below, the voltmeter reading is zero.

When the temperature of the thermistor T is increased, the voltmeter reading changes.

Which change to the circuit will restore the voltmeter to zero?

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 ?

In the circuit below, the initial voltmeter reading is zero.

The temperature of the negative temperature coefficient thermistor T is then increased.

Which change to the circuit could restore the voltmeter reading to zero?

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 student carries out an experiment to determine the resistivity of a metal wire. She determines the resistance from measurements of potential difference between the ends of the wire and the corresponding current. She measures the length of the wire with a ruler and the diameter of the wire using a micrometer. Each measurement is made with an uncertainty of 1%

Which measurement gives the largest uncertainty in the calculated value of the resistivity?

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 resistance of wire P is R.

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