Discharging a Capacitor (CIE A Level Physics)

The variation with potential difference V of the charge Q on one of the plates of a capacitor is shown in Fig. 1.1.

Fig. 1.1

The capacitor is connected to a 12.0 V power supply and two resistors P and Q are shown in Fig. 1.2.

Fig. 1.2

The resistance of P is 28 kΩ and the resistance of Q is 280 kΩ.

The switch can be in either position A or position B.

The switch is in position A so that the capacitor is fully charged.

Calculate the energy E stored in the capacitor.

The switch is now moved to position B.

 Show that the time constant of the charge circuit is 4.2 s.

The fully charged capacitor in (a) stores energy E. 

Determine the time t taken for the stored energy to decrease from E to E/4.

A second identical capacitor is connected in series with the first capacitor. 

State and explain the change, if any, to the time constant of the discharge circuit.

A capacitor of capacitance 520 μF is connected to a battery of electromotive force (e.m.f.) 36 V in the circuit of Fig. 1.1.

Fig. 1.1

The two-way switch is initially at position A.

X and Y are identical long straight wires, each with a resistance of 2.4 kΩ. These wires are parallel to each other. Wire Y is connected to a voltmeter. 

At time t = 0, switch S is moved to position B so the capacitor discharges through wire X.

Calculate the time constant τ of the discharge circuit.

Fig 1.3

[1]

Explain why there is an induced e.m.f. across wire Y during the discharge of the capacitor.

A 590 nF capacitor is charged fully from a 20 V battery. At time t = 0 the capacitor begins to discharge through a resistor. When t = 15 s the energy remaining in the capacitor is one eighth of the energy it stored at 20 V. 

Show that the potential difference across the capacitor when t = 15 s is around 7 V. 

Calculate the time constant of this circuit.

Calculate the resistance of the fixed resistor in MΩ.

A signal generator is used to produce a square wave, which is an oscillating voltage between 0 V and some maximum. It is connected to an oscilloscope, and the waveform generated is shown in Fig. 1.1.

The Y-voltage gain is set to 2 V div^–1 and the time-base is set to 0.5 ms. 

Fig. 1.1

The signal generator and oscilloscope are then connected to a capacitor C and resistor R, which has a resistance of 2.2 kΩ, as shown in Fig. 1.1. 

Fig. 1.2

Sketch the waveform shown by the oscilloscope over one complete period. You may assume the capacitor can fully charge or discharge over the course of half a period of the signal. 

Describe how the waveform displayed on the oscilloscope in Fig. 1.1 could be used to determine the capacitance of the capacitor C. 

Where appropriate, include calculations in your response.

Discuss how the settings on the oscilloscope could be adjusted to reduce the uncertainty in determining the time constant. 

The circuit in Fig. 1.2 is now adjusted by placing an identical resistor in parallel with R. This changes the waveform displayed by the oscilloscope. 

Discuss the effects on the circuit and on the shape of the waveform displayed on the capacitor following the adjustment made to the circuit. You may wish to include a sketch with your answer.

Pacemakers are devices which are used to deliver short pulses of charge to a patient who is suffering from a cardiac arrest. 

Fig. 1.1 shows a circuit that is used in such a pacemaker. 

The capacitor has a capacitance of 20 μF and is connected to a resistor of 220 kΩ. This is connected to a power supply, but upon changing a two-way switch it forms a circuit with heart tissue. This has a resistance of 400 Ω.

Fig. 1.1

The capacitor is fully charged when at t = 0 s, then the two-way switch connects it to the heart tissue. 

Fig. 1.2

Sketch the graph on the axes provided in Fig. 1.2 that shows how the voltage across the capacitor varies over 10 seconds. 

Show that, when charging from 0 C, the time taken for the charge stored in the capacitor to reach 5.0 mC is approximately 0.3 s.

The equation for the charging of a capacitor is:

where  represents the maximum value of a given quantity.

Pacemaker manufacturers need efficient ways of measuring how quickly the charge and energy are released by capacitors. 

One approach uses the ‘exponential factor’ σ , which is a ratio of the energy remaining in a capacitor to its initial energy stored.

Show that the exponential factor σ is given by: