Capacitors and capacitance

An electroscope is an instrument used to measure electric charge.

The diagram shows one type of electroscope.

The diagram below shows the needle of the electroscope when it is negatively charged. The centre of mass, X, of the needle has been marked on the diagram.

Suggest why the centre of mass of the needle is a short distance below the pivot.

A student charges the electroscope by connecting the metal plate to the negative terminal of a high voltage power supply.

They record the deflection angle of the needle for a range of values of the voltage measured using a voltmeter.

Their results are shown in the graph below.

The student concludes that the voltmeter has a zero error of 300 V.

Explain their reasoning.

The capacitance of the electroscope is 20 pF.

Estimate the number of electrons stored on the electroscope when the deflection angle is 45°.

This question is about lightning.

Sheet lightning occurs when there is an electrical discharge between the upper and lower regions of a thunder cloud. The upper regions are positive and the lower regions are negative.

The thunder cloud can be modelled as an ideal parallel plate capacitor with circular horizontal plates.

The data for the capacitor comes from the cloud.

The diagram shows the plates of the model capacitor superimposed on the cloud.

Draw on the diagram to show the electric field lines between capacitor plates.

Suggest why the actual electric field lines of the cloud would differ from what you have drawn.

Calculate the capacitance C of the model capacitor.

Assume the permittivity of the material of the cloud is the same as the permittivity of free space.

Calculate the magnitude of the charge Q on one of the plates of the model capacitor.

Fork lightning is an electrical discharge that occurs between the bottom of the cloud and the surface of the Earth.

Another cloud has a charge of 155 C and is at a height of 2.0 km.

The surface of the Earth has an electrical potential V of 0 V.

Assume the cloud acts as a point charge.

Calculate the magnitude of the electrical potential V between the cloud and the surface of the Earth.

A fork lightning strike has a duration of 25 ms. The cloud discharges at a constant rate. The cloud is uncharged after the strike.

Calculate the number of electrons reaching the ground in 1.0 ms.

A group of students investigate the circuit shown in the figure below.

There are N capacitors, each of capacitance C, connected in parallel.

There are N resistors, each of resistance R, connected in series.

Initially, the students close the switch X. They then note the reading on the ammeter.

The students then open the switch. They record the time T for the reading on the ammeter to fall to half of its initial value.

The table below shows the students’ results.

Show that T = (ln 2)N²RC. (a)

The results table is repeated below.

Complete the last column for N = 1 and N = 2 in the table below.

The students begin to plot a graph of T (y-axis) against N² (x-axis). (d)

Draw a straight line of best fit on the graph.

Calculate the gradient of the straight line of best fit.

The value of C is known to be 1000 F ± 5%.

Use your gradient value from (iii) to find a value for R, in units of k , including an absolute uncertainty.

R =........ ±........ k

Following the investigation, the students discovered that the sixth 1000 F capacitor connected to the circuit was actually two 470 F capacitors connected in parallel.

State the type of error caused by this mistake.

Explain the effect that this error would have had on the calculated value of R.

Large power stations generate an electrical power of about 1 GW.

Current methods of energy production that use nuclear fusion are unable to produce enough energy for large-scale energy production. A proposed method of controlling nuclear fusion is inertial confinement fusion (ICF). ICF uses a large number of powerful lasers to create the high temperatures required for nuclear fusion to occur.

One ICF experiment uses a network of capacitors to store the energy needed to power the lasers. When the network is fully charged:

• potential difference across the network = 24 kV

• total energy stored in the network = 400 MJ

Calculate the total capacitance, C, of the network.

Explain why the individual capacitors in the network should be connected in parallel in order to produce this total capacitance.

The total stored energy must be released in a time of less than 1 millisecond.

Explain, using a calculation, why the lasers are powered by the network of capacitors instead of being connected directly to the mains electricity supply.

The fusion reaction in the ICF experiment is

Calculate the number of fusion reactions that must occur for the energy released by fusion to be equal to the electrical energy stored in the network of capacitors.

• mass of deuterium = 2.014102 u

• mass of tritium = 3.016049 u

• mass of alpha particle = 4.002603 u

• mass of neutron = 1.008665 u

The student decides to plot T against R to obtain a straight-line graph.

Show that the line has gradient = C ln2.

The data points plotted by the student are shown below.

1 Draw a best-fit straight line through the data points and use the gradient of this line to determine C.

2 Use your answer in (ii)1 to calculate the permittivity of the paper.

=........ F m^–1

The diagram below shows a circuit containing two capacitors which are both initially uncharged. The battery has e.m.f. E and negligible internal resistance.

The switch S is first moved to position A until the capacitor of capacitance C₀ is fully charged.

The switch S is then moved to position B. The initial charge stored by the capacitor of capacitance C₀ is shared between the two capacitors. The final reading on the voltmeter is V.

A student wants to determine the values of E and C₀ by repeating the experiment above and measuring the potential difference (p.d.) V for a selection of capacitors of capacitance C. The student decides to plot a graph of against C.

The data points, error bars and the line of best fit drawn by the student are shown in the graph below.

The gradient of the line of best fit is 51 V^–1 F^–1. The value of E is 9.1 V.

Determine the value of C₀ in millifarads (mF). Write your answer to 2 significant figures.

Draw on the graph a straight line of worst fit.

Use this line to determine the absolute uncertainty in your value of C₀. Write your answer to an appropriate number of significant figures.

Use Fig. 22.2 to propose and carry out a test to show that Q is inversely proportional to d.

Test proposed:

Working:

Use capacitor equations to show that Q is inversely proportional to d.

* Fig. 22.4 shows an arrangement used by a student to investigate the forces experienced by a small length of charged gold foil placed in a uniform electric field.

The two vertical metal plates are connected to a high-voltage supply.

The foil is given a positive charge by briefly touching it to the positive plate.

The angle made with the vertical by the foil in the electric field is given by the expression

where q is the charge on the foil, E is the electric field strength between the plates and W the weight of the foil.

The angle can be determined by taking photographs with the camera of a mobile phone.

Describe how the student can safely conduct an experiment to investigate the relationship between and E.

Identify any variables that must be controlled.

Additional answer space if required.

An isolated metal sphere is charged using a power supply.

Which single quantity can be used to determine the capacitance of the sphere?

A The diameter of the sphere.

B The charge on the sphere.

C The resistance of the metal.

D The e.m.f. of the power supply.

Your answer

The graph below shows the variation of potential difference V with charge Q for a capacitor.

Which row is correct for the gradient of the graph and the area under the graph?

Your answer

A capacitor of capacitance 7.2 pF consists of two parallel metal plates separated by an insulator of thickness 1.2 mm. The area of overlap between the plates is 4.0 10⁻⁴ m². Calculate the permittivity of the insulator between the capacitor plates.

permittivity =........ F m⁻¹

Fig. 21 shows a circuit.

The capacitance of each capacitor is 1000 F. The resistance of the resistor is 10 k. The cell has e.m.f. 1.5 V and negligible internal resistance.

Calculate the total capacitance C in the circuit.

C =........ F

The switch S is closed at time t = 0. There is zero potential difference across the capacitors at t = 0. Calculate the potential difference V across the resistor at time t = 12 s.