Charging and discharging capacitors

A material used between the plates of a capacitor has a dielectric constant of 7.0

State what is meant by a dielectric constant of 7.0

Figure 3 shows a circuit that is used to charge and discharge a capacitor.

The battery has negligible internal resistance.

The two resistors in the circuit are identical.

The capacitor is fully charged and the switch is then moved from position A to position B at time t = 0

Figure 4 shows the variation of the microammeter reading with t until t = 120 s.

The time constant of the discharging circuit is T.

Determine T using Figure 4.

Electronic engineers consider a capacitor to be fully discharged when t = 5 T.

The capacitor is fully discharged and then the switch is moved from position B to position A as shown in Figure 3.

Draw, on Figure 5, the variation of the microammeter reading with time after the switch is moved to position A. In Figure 5, the time is 0 at the instant when the switch is moved to position A.

A capacitor is initially uncharged and is then charged using a constant current. This is achieved by reducing the resistance of a variable resistor R as the charge on the capacitor increases.

The capacitor has a capacitance of 300 mF and is fully charged in a time t = 100 s. The cell has an emf of 2.0 V and negligible internal resistance. The charging current is 6.0 mA.

What is the resistance of R at t = 30 s ?

Figure 1 shows a circuit used to charge capacitor C. The battery has negligible internal resistance.

The capacitance of C is known.

The switch is closed at time t = 0 and the potential difference V_C across C is recorded at different times t. Figure 2 shows the variation of V_C with t.

Explain how a gradient of the graph in Figure 2 can be used to determine the initial

The potential difference V_R across R is also recorded. Figure 3 shows the variation of V_R with t between t = 20 s and t = 45 s.

The capacitance of C is 31.0 μF.

Determine, using Figure 3, the time constant of the circuit. Go on to show that the resistance of R is about 2.4 × 10⁵ Ω.

The current I₀ at time t = 0 is 3.6 × 10⁻⁵ A.

Determine the time at which V_C is 6.0 V.

Figure 8 shows a circuit used to investigate the charge and discharge of a capacitor of capacitance C using resistors of resistances R₁ and R₂.

The battery has an emf of 6.0 V and negligible internal resistance.

Show that the time taken for the capacitor to charge from 2.0 V to 4.0 V is

The capacitor is fully discharged. The capacitor is then charged until the potential difference (pd) across it is 4.0 V. Figure 9 shows the variation with time of the ammeter reading as the capacitor is charged.

Show that the capacitance of the capacitor is about 1 × 10⁻⁴ F.

When the pd reaches 4.0 V the switch is immediately set to discharge the capacitor. When the pd reaches 2.0 V the switch is immediately set to charge the capacitor.

Figure 10 shows how the pd across the capacitor varies with time.

A 30 μF capacitor is charged by connecting it to a battery of emf 4.0 V. The initial charge on the capacitor is Q₀.

The capacitor is then discharged through a 500 kΩ resistor. The time constant for the circuit is Τ.

Which is correct?

A fully charged capacitor of capacitance 2.0 mF discharges through a 15 kΩ resistor.

What fraction of the stored energy remains after 1.0 minute?

The capacitor in the circuit is initially uncharged. The switch is closed at time t = 0

Which pair of graphs shows how the potential difference V across the capacitor and the current I in the circuit change with time t ?

An analogue voltmeter has a resistance that is much less than that of a modern digital voltmeter. Analogue meters can be damaged if the full-scale reading is exceeded. Figure 6 shows a dual-range analogue voltmeter with a zero error.

The voltmeter is set to the more sensitive range and then used in a circuit.

What is the potential difference (pd) between the terminals of the voltmeter when a full-scale reading is indicated?

Tick () one box.

Explain the use of the mirror when reading the meter.

A student corrects the zero error on the meter and then assembles the circuit shown in Figure 7. The capacitance of the capacitor C is not known.

The output pd of the power supply is set to zero. The student connects the flying lead to socket X and adjusts the output pd until the voltmeter reading is full scale (15 V). She disconnects the flying lead from socket X so that C discharges through the voltmeter.

She measures the time T ½ for the voltmeter reading V to fall from 10 V to 5 V.

She repeats this process several times.

Table 1 shows the student’s results, none of which is anomalous.

Determine the percentage uncertainty in T ½.

Show that the time constant for the discharge circuit is about 17 s.

The student thinks that the time constant of the circuit in Figure 7 is directly proportional to the range of the meter. To test her theory, she repeats the experiment with the voltmeter set to the 3 V range. She expects T ½ to be about 2.5 s.

Explain:

• what the student should do, before connecting capacitor C to the 0 V and 3 V sockets, to avoid exceeding the full-scale reading on the voltmeter

• how she should develop her procedure to get an accurate result for the time constant

• how she should use her result to check whether her theory is correct.

The student wants to find the resistance of the voltmeter when it is set to the 15 V range. She replaces C with an 820 µF capacitor and charges it to 15 V. She discharges the capacitor through the voltmeter, starting a stopwatch when V is 14 V. She records the stopwatch reading t at other values of V as the capacitor discharges.

Table 2 shows her results.

Suggest two reasons why the student selected the values of V shown in Table 2. Explain each of your answers.

Figure 8 shows a graph of the experimental data.

Show, using Figure 8, that the resistance of the voltmeter is about 16 kΩ.

Determine the current in the voltmeter at t = 10 s.

Figure 6 shows an oscilloscope connected across resistor R which is in series with an ac supply. The supply provides a sinusoidal output of peak voltage 15 V.

Calculate the rms voltage of the supply.

Figure 7 shows the trace of the waveform displayed on the oscilloscope.

Determine the y-voltage gain of the oscilloscope used for Figure 7.

A dc supply gives the same rate of energy dissipation in R as the ac supply in Figure 6.

Draw the trace of the output of the dc supply on Figure 7. The oscilloscope settings remain the same.

The ac supply shown in Figure 6 is replaced with a square-wave generator operating between 0 and +15 V.

Figure 8 shows the trace of the new waveform displayed on the oscilloscope. The time-base is set to 5.0 × 10⁻⁴ s div⁻¹.

Calculate the frequency of the square waves.

Figure 9 shows the arrangement with the square-wave generator connected to an RC circuit. A capacitor C is placed in series with the resistor R. The oscilloscope is connected across the capacitor C.

The capacitor charges and discharges.

Figure 10 shows the trace of the waveform displayed on the oscilloscope. The settings of the oscilloscope remain the same as in Question 04.4.

Deduce the time constant for the RC circuit, explaining each step of your method.

State and explain a change to one control setting on the oscilloscope that would reduce the uncertainty in the value of the time constant.

A 10 μF capacitor stores 4.5 mJ of energy. It then discharges through a 25 Ω resistor.

What is the maximum current during the discharge of the capacitor?

A 1.0 µF capacitor initially stores 15 µC of charge. It then discharges through a 25 Ω resistor.

What is the maximum current during the discharge of the capacitor?

The initial potential difference across a capacitor is V₀. The capacitor discharges through a circuit of time constant T. The base of natural logarithms is e.

What is the potential difference across the capacitor after time T ?

State what is meant by a capacitance of 370 F

The charging of a 370 F capacitor is investigated using the circuit shown in Figure 4 Both meters in the circuit are ideal.

The power supply of emf 9.8 V has a negligible internal resistance. The capacitor is initially uncharged. When the switch is closed at time t = 0 charge begins to flow through resistor R. The time constant of the charging circuit is 1.0 s

Calculate the resistance of R.

Identify, with the symbol X on Figure 5, the potential difference (pd) across the capacitor when the switch has been closed for 2.0 s Sketch the graph that shows how the pd varies from t = 0 to t = 2.0 s

Calculate the time taken for the charging current to fall to half its initial value.

Calculate the time taken for the charge on the capacitor to reach 3.0 mC

What fraction of the original charge remains on the capacitor 4.8 s after the discharge begins?

The Y-voltage gain and time-base settings of the oscilloscope are shown in Figure 6.

When switch S is open (off) the oscilloscope displays the waveform shown in Figure 7.

When S is closed (on) the oscilloscope displays the waveform shown in Figure 8.

Determine the peak-to-peak voltage V of the waveform shown in Figure 8.

Determine the frequency f of the waveform shown in Figure 8.

Figure 9 shows the signal generator connected in series with a resistor R and a capacitor C.

The oscilloscope is connected across the capacitor. The Y-voltage gain and time-base settings are still the same as shown in Figure 6.

When S is closed (on) the oscilloscope displays the waveform shown in Figure 10.

Determine the time constant of the circuit in Figure 9.

A student suggests that setting the time-base to 0.2 ms division might reduce uncertainty in the determination of the time constant.

State and explain any possible advantage or disadvantage in making this suggested adjustment.

The student connects an identical resistor in parallel with R and uses the oscilloscope to display the waveform across C.

Draw on Figure 11 the waveform you expect the student to see.

The waveform of Figure 10 is shown as a dashed line to help you show how the waveform changes.

Explain the change in the waveform.

Figure 12a is a graph of voltage against time showing the output of the signal generator. Figure 12b shows the voltage across C during the same time interval. The student interchanges the positions of R and C and connects the oscilloscope across R. Complete Figure 12c to draw the voltage across R during the time interval.

State and explain what changes, if any, the student needs to make to the settings of the oscilloscope so the waveform across R is fully displayed.