7.6 Capacitance

An air-filled parallel-plate capacitor is charged from a source of e.m.f. The electric field has a strength E between the plates. The capacitor is disconnected from the source of e.m.f. and the separation between the isolated plates is doubled. 

State the final value of electric field strength between the plates and explain your answer.

The capacitor consists of two parallel square plates of side separated by distance d. The capacitor contains air as a dielectric. The capacitance of the arrangement is C. 

Determine the capacitance of a second capacitor with square plates of side 2 separated by air at a distance of  .

The first capacitor is then filled with a dielectric which increases its capacitance to 2C. 

The second capacitor still has air as its dielectric. 

Determine the relative permittivity of the dielectric in the first capacitor.

One of the capacitors is a variable capacitor which can be varied from 8.0 to 4.0 × 10^–12 F. 

The capacitor is set to its maximum capacitance and fully charged using a 36 V supply. The capacitor is then disconnected from the supply and isolated. Finally, the capacitance is reduced to its minimum value without any charge being lost by the capacitor. 

Determine the potential difference (p.d.) across the capacitor and the charge it stores after the capacitance has been reduced.

Dennis is an electrical engineer and wants to investigate how capacitors store and release charge and energy. 

He firstly designs a circuit to determine the capacitance C of a capacitor. This circuit is shown in Figure 1. The switch S is held in position A until the capacitor is fully charged to
6 V and is then moved to position B so that it fully discharges through the microammeter and the variable resistor R.

Figure 1

While discharging, Dennis continuously adjusts the variable resistor R to keep the current constant until the capacitor is fully discharged. Figure 2 shows how the current I varies during the discharging process. 

Figure 2

Calculate the capacitance of the capacitor.

Dennis understands that as the capacitor discharges, it releases both charge and energy. 

Sketch graphs on the axes provided to show how the following quantities vary with time as the capacitor discharges: 

1. Charge
2. Energy 

You should include all important labels as appropriate.

Dennis disconnects the circuit and increases the EMF of the power supply by 50%. 

He states that by doing this, the charge stored in the capacitor will increase by 50% and the energy stored in the capacitor will increase by 125%. 

Justify Dennis’s statement with suitable calculations.

The insulating material between capacitor plates is called the dielectric. 

Figure 1 shows two capacitor circuits, both connected to the same magnitude of EMF, , One of the capacitors A has a dielectric between its plates. The other capacitor, B has a vacuum between its plates. Their dimensions are otherwise identical. 

Figure 1

Discuss the advantages of including a dielectric in a capacitor, with reference to the electric field between the plates. 

The presence of a dielectric makes the capacitance of capacitor A,  , 50% greater than the capacitance of capacitor B, . 

Show that the relative permittivity of the dielectric in capacitor A is 1.5.

Both capacitors are connected as shown in Figure 1 and fully charged. The energy stored in capacitor A is 10 J. 

Calculate the amount of energy stored in capacitor B.

A capacitor with capacitance C = 220 μF is attached to an AC voltage source which provides a voltage of sin (ωt) as shown in Figure 1. This circuit is called ‘purely capacitive’, because we assume there is no resistance in it. 

Figure 1

The power supply is adjusted such that  = 6.0 V and ω = 4000 rad s^–1­. 

Calculate the magnitude of the current flowing in the circuit at time t = 3.14 s. 

You may wish to use the following mathematical result: 

          sin (ωt) = ω cos (ωt) 

where the operation  can be thought of as a rate of change.

The variation of voltage V and current I in the circuit is shown in Figure 2. 

Figure 2

Discuss the phase difference between the variation of V and I in the circuit.

Capacitive reactance X is the opposition to current flow in a purely capacitive circuit, like the one described in Figure 1. It can be thought of as similar to resistance, in that it is measured by the same units Ω. 

By considering the ratio of the maximum voltage and current in the circuit, show that the capacitive reactance X is given by: 

      X = 

and verify that X has the same units as resistance.

A capacitorof capacitance 450pF is made up of two parallel metal plates. Each plate has an area of 150cm². A sheet of dielectric material with a relative permittivity of 3.2 fills the gap between the two parallel plates. 

Calculate the distance between the two plates.

The charge on the capacitor is 28 nC. The dielectric material is removed without discharging the capacitor. The separation of the plates remains unchanged. 

Show that the potential difference between the plates is about 200V. 

Relative permittivity of air = 1.0

Calculate the energy stored by the capacitor

When the dielectric material is removed from between the plates there is an increase in energy. 

Explain this observation.

The surface of the Earth and the base of a charged cloud can be treated as a parallel plate capacitor.

Calculate the capacitance of an Earth-cloud system if the cloud is 1200m above the surface of the earth and the cloud has a base area of 1.8 ×10⁶m².

Relative permittivity of air = 1.0

If a potential difference of 7 × 10⁵ V exists across each metre of air, the cloud will discharge across the air to the Earth’s surface.

Show that the potential difference between the cloud and the Earth is 840MV.

Calculate the maximum amount of energy which can be stored in the charged Earth-cloud system.

Negative charge gathers at the bottom of the cloud before discharging in the form of a lightning strike.

Assuming the negative charge to be electrons, calculate:

An uncharged 3.5 nF capacitor is connected to a 2.0 V power supply and becomes fully charged. 

How many electrons are transferred to the negative plate of the capacitor during this charging process?

State and explain how many electrons are transferred from the positive plate to the power supply. 

A dielectric is inserted between the capacitor plates, which acts to increase its capacitance. This is shown in Figure 1. 

Figure 1

Draw lines on Figure 1 to indicate the direction of the electric field due to:

Using your answer to part (c), explain why the dielectric increases the capacitance of the capacitor. 

Figure 1 shows the discharge of a capacitor in a heart pacemaker. 

Figure 1

Explain how the rate of change of potential difference changes as the capacitor discharges.

The capacitor has a capacitance of 140μF. 

Calculate the energy lost by the capacitor whilst it discharges during the first 30ms.

Give your answer as a percentage of the initial energy.

The charge which is lost as the capacitor discharges is used by the pacemaker to produce a single pulse for the heart. 

Calculate the charge, in μC,of a single pulse.

The pacemaker has been designed to operate for a minimum of 3 years, delivering 60 constant pulses per minute. 

Calculate the minimum charge of the power supply if it is to operate over the 3 years. 

Give your answer in Amp-hours.