7.7 Capacitor Charge & Discharge

Free falling bodies can be used to measure the gravitational field strength, g. Typically, this requires measurements of distance and time. 

A lab technician wants to determine if more accurate measurements of g can be achieved by using a capacitor instead of a stopwatch. She invents a setup using a steel ball bearing, which is connected to a capacitor and power source using steel contacts. 

The capacitoris fully charged to a voltage  after which the ball bearing is released. As it falls, the capacitor discharges through a resistor, until the ball bearing collides with a trap door which breaks the circuit. The voltage across the capacitor at this instant is V.  

Figure 1 and Figure 2 show a front and side view of the setup. 

    Figure 1: front view                                                Figure 2: side view

Show that the distance travelled by the ball bearing, s is given by: 

         s = 8g (ln)²

Suggest how the lab technician should carry out the experiment to determine a value for g. 

Include reference to the independent and dependent variables, any data processing required, and how the data should be appropriately analysed.

The technician tests the set up and is satisfied that all works as expected, with an initial voltage of 10.00 V and a final voltage of 8.74 V as the ball bearing collides with the trap door. 

She states that there is a smaller uncertainty in the value of g from this experimental method than the traditional way of measuring displacement s and time t directly using a stopwatch.

Discuss the conclusion of the technician by comparing the sources of uncertainty in both experimental methods.

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

Figure 1 shows a circuit  that is used in such a pacemaker. 

Figure 1

The capacitor is initially uncharged when at t = 0 s, then the two-way switch connects it to a power supply of 4.0 kV. 

Sketch the graph on the axes provided in Figure 2 that shows how the voltage across the capacitor varies over 10 seconds. 

Figure 2

Show that the time taken for the charge stored in the capacitor to reach 5.0 mC is approximately 0.3 s.  

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 E to its initial energy stored  . For a given time interval, a smaller ratio indicates a quicker greater discharge of energy. 

Show that the exponential factor σ is given by:

            σ =

A student sets up an experiment to investigate energy transfers between a projectile and a capacitor. 

They arrange the apparatus so that, if they flick a ball bearing of mass m = 40 g from the top of a bench, it collides with the centre of two pieces of metal foil which are 0.50 m apart and connected in an electric circuit as shown in Figure 1. 

As the ball bearing collides with the first metal strip, the capacitor begins to discharge through the resistor. As the ball bearing collides with the second metal strip, the capacitor stops discharging.  

Figure 1

The results of the experiment are shown in Table 1: 

Table 1

Using the information given in Figure 1 and Table 1, calculate the horizontal speed of the ball bearing.

As the ball is struck, you may assume the vertical descent between the bench and the first metal strip is negligible. 

Show that the ball bearing transfers approximately 100 times more potential energy than the capacitor in the time taken for it to discharge.

The capacitor in Figure 1 is replaced with an unmarked capacitor, and the experiment is repeated for one additional trial.

 The results of this trial are included in Table 2: 

Table 2

Use the data in Table 2 to determine the capacitance of the unmarked capacitor and comment on whether your answer is as expected.

A capacitor is charged to a maximum of 13.2 µC with a time constant of 15 ms. 

Calculate the charge stored in the capacitor after 12 ms.

The graph in Figure 1 shows how the potential difference (p.d.) V varies with the charge Q stored by a different capacitor as V is increased from 9.0 V to 12.0 V. 

Figure 1

Use Figure 1 to determine an accurate value for the capacitance of this capacitor.

When the capacitor is discharged through a fixed resistor R, the charge decreases by 90% in 35 s. 

Calculate the resistance of R in kΩ.

Figure 1 shows a circuit consisting of a resistor and a capacitor of capacitance 7.1 µF. 

 Figure 1

Switch  is closed and switch  is left open. 

Calculate the charge stored by the capacitor.

Switch  is now opened and switch  is closed. 

Describe and explain in terms of the movement of electrons how the potential difference across the capacitor changes.

The time constant of the circuit when the capacitor is discharging is 5.0 minutes. 

Figure 2

Sketch a graph on Figure 2 to show how the potential across the fixed resistor varies with time for the first 12 minutes after switch  closes.

Calculate the initial current in the circuit when the capacitor begins to discharge.

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 time constant of this circuit.

Calculate the resistance of the fixed resistor in MΩ.

A student decides to confirm the value of the capacitance of this capacitor by determining the time constant of the circuit when the capacitor discharges through a fixed resistor. 

Describe an experiment to do this. Include in your answer: 

• A circuit diagram.
• An outline of a procedure.
• An explanation of how you would use the data to determine the time constant.