Charging and discharging capacitors — OCR A-Level Physics
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Charging and discharging capacitors explained
This row is a specification heading, not an assessed statement, so it signals the start of the charging and discharging topic rather than a question target.
Read the full explanation
Read it as a signpost: the following sub-statements cover charging and discharging through a resistor, and the techniques and procedures used to investigate charge and discharge with meters and data-loggers. When studying, build the qualitative picture first: closing a switch connects a capacitor to a supply through a resistor, current is largest at the start and falls as the capacitor charges, and the potential difference across the capacitor rises towards the supply value. Discharging reverses this, with current flowing the other way and falling towards zero. Then attach the equations and graphs to that picture.
(i) charging and discharging capacitor through a resistor
When a capacitor charges through a resistor, the current is initially V/R and falls exponentially as the capacitor voltage rises towards the supply value. Discharging reverses the process: the capacitor drives current back through the resistor, and charge, current and voltage all decay exponentially. The time constant τ = RC is the time for charge or current to fall to 1/e of its starting value, about 37%, and for a discharging capacitor q = Q₀e^(−t/RC). For example, a 100 µF capacitor with a 10 kΩ resistor has τ = 100 × 10⁻⁶ F × 10 × 10³ Ω = 1.0 s. Graphs of charge, current and potential difference against time are exponential curves, and the area under a current-time graph gives charge.
(ii) techniques and procedures to investigate the charge and the discharge of a capacitor using both meters and data-loggers
To investigate charging and discharging, connect the capacitor, resistor and switch in series with a suitable supply, then measure either the current through the circuit or the potential difference across the capacitor. With meters, a voltmeter across the capacitor and an ammeter in series let you take readings at timed intervals, but manual timing is imprecise because the change is fast at the start. A data-logger with a voltage or current sensor records many readings per second automatically, giving smooth exponential graphs and allowing the time constant to be found from the curve. Choose a resistor and capacitor so τ = RC is long enough to observe, and use a sensor range that matches the expected voltages.
(b) time constant of a capacitor–resistor circuit; τ = CR
The time constant τ of a capacitor–resistor circuit is defined by τ = CR, where C is capacitance in farads (F) and R is resistance in ohms (Ω). Because 1 F × 1 Ω = 1 s, τ has units of seconds. For example, a 470 μF capacitor (470 × 10⁻⁶ F) with a 10 kΩ resistor (10 × 10³ Ω) gives τ = (470 × 10⁻⁶) × (10 × 10³) = 4.7 s. In a discharging circuit, τ is the time for the charge, potential difference or current to fall to 1/e (about 37%) of its initial value. In a charging circuit, τ is the time to reach (1 − 1/e) (about 63%) of the final value. τ is not the time to fully charge or discharge; after each interval of τ the remaining change is reduced by a factor of 1/e, so the process is asymptotic. The time constant depends only on C and R, not on the initial charge or the applied voltage.
(c) equations of the form x = x₀e^(−t/CR) and x = x₀(1 − e^(−t/CR)) for capacitor–resistor circuits
In a capacitor–resistor circuit, the time constant \(\tau = CR\) determines the rate of charging and discharging. For a discharging capacitor, charge \(Q\), potential difference \(V\), and current \(I\) all decay exponentially according to \(x = x_0 e^{-t/CR}\), where \(x_0\) is the initial value. During charging, the charge and potential difference rise towards a maximum steady value according to \(x = x_0(1 - e^{-t/CR})\). However, the charging current does not rise; it decays exponentially from an initial maximum as the capacitor opposes further charge flow, following \(I = I_0 e^{-t/CR}\). The exponent \(-t/CR\) must be dimensionless, requiring \(t\) and \(CR\) to be in seconds. For example, after \(t = CR\), a discharging quantity falls to \(x_0 e^{-1} \approx 0.37 x_0\), while a charging \(V\) or \(Q\) rises to \(x_0(1 - e^{-1}) \approx 0.63 x_0\).
(d) graphical methods and spreadsheet modelling of the equation ΔQ/Δt = −Q/CR for a discharging capacitor
The differential equation ΔQ/Δt = −Q/CR describes the rate of change of charge Q on a discharging capacitor, where C is capacitance, R is resistance and CR is the time constant τ. The negative sign shows that charge is decreasing. Graphical methods include plotting Q against t to obtain an exponential decay curve, or plotting ln Q against t to obtain a straight line of gradient −1/CR. The gradient of the Q–t graph at any instant equals ΔQ/Δt, so the initial gradient is −Q₀/CR. Spreadsheet modelling involves using a small time step Δt to update Q iteratively: Q_new = Q_old + (ΔQ/Δt) × Δt, where ΔQ/Δt is calculated from the current Q. This numerical method approximates the exponential solution and allows the effect of changing C or R to be explored. The accuracy improves as Δt is made smaller.
(e) exponential decay graph; constant-ratio property of such a graph.
An exponential decay graph shows a quantity decreasing at a rate proportional to its current value, producing a curve that asymptotically approaches zero. For a discharging capacitor, the charge Q, potential difference V or current I follows Q = Q₀e^(−t/CR). A key property is the constant-ratio property: for any equal time intervals, the ratio of the quantity at the end of the interval to that at the start is constant. For example, if the time interval equals the time constant τ = CR, the ratio is e^(−1) ≈ 0.37. If the interval is 2τ, the ratio is e^(−2) ≈ 0.14. This property allows identification of exponential decay from experimental data: successive values at equal time intervals should have a constant ratio. It also means that a graph of ln Q against t is a straight line with gradient −1/CR.
Your focus
- Identify that this heading introduces charging and discharging of capacitors.
- Describe qualitatively how charge, current and potential difference change with time.
- Locate and study the sub-statements on resistor circuits and investigation techniques.
Show all 21 objectives
- Describe how current and potential difference change during charging and discharging through a resistor.
- Calculate and interpret the time constant of an RC circuit.
- Apply exponential equations to find charge or current at a given time.
- Describe how to set up a circuit to charge and discharge a capacitor.
- Compare meter readings with data-logger measurements for this investigation.
- Select appropriate component values and sensor ranges to obtain usable data.
- State and use the equation τ = CR to calculate the time constant of a capacitor–resistor circuit.
- Explain the physical meaning of the time constant in terms of the 1/e and 1 − 1/e points on charging and discharging graphs.
- Convert between microfarads, nanofarads and farads, and between kilohms, megohms and ohms, when performing time-constant calculations.
- Apply \(x = x_0 e^{-t/CR}\) to calculate decaying charge, p.d., and current in discharging circuits, and current in charging circuits.
- Apply \(x = x_0(1 - e^{-t/CR})\) to calculate rising charge and p.d. in charging circuits.
- Verify that the exponent in the equations is dimensionless by checking units.
- Interpret and use the differential equation ΔQ/Δt = −Q/CR for a discharging capacitor.
- Determine the time constant from the gradient of a graph of ln Q against t or from the initial gradient of a Q–t graph.
- Describe how a spreadsheet can be used to model the discharging process and explain how the choice of time step affects accuracy.
- Describe the shape of an exponential decay graph and explain its asymptotic approach to zero.
- Apply the constant-ratio property to identify exponential decay and to calculate the time constant from data.
- Use a logarithmic plot to confirm exponential decay and determine the time constant from the gradient.
Charging and discharging capacitors exam tips
Marking Points
- States that charging current is largest at the start and falls exponentially as the capacitor voltage rises.
- Applies the time constant τ = RC and interprets it as the time to fall to 1/e of the initial value.
- Uses q = Q₀e^(−t/RC) or I = I₀e^(−t/RC) for discharging.
- Explains that the capacitor voltage rises towards the supply voltage during charging and falls towards zero during discharging.
- Relates the area under a current-time graph to charge transferred.
- Describes a series circuit containing a capacitor, resistor, switch and supply for charging and discharging.
- States that a voltmeter is connected in parallel with the capacitor and an ammeter in series with the circuit.
- Explains that a data-logger records many readings automatically and produces a smooth graph.
- Identifies that manual meter readings at timed intervals are less precise because the initial change is rapid.
- Chooses component values so the time constant is long enough to measure, and selects a suitable sensor range.
- State the defining equation τ = CR and identify C as capacitance in farads and R as resistance in ohms.
- Show that the product CR has units of seconds, since 1 F × 1 Ω = 1 s.
- Explain that in a discharging circuit τ is the time for charge, potential difference or current to fall to 1/e of its initial value (about 37%).
- Explain that in a charging circuit τ is the time to reach (1 − 1/e) of the final value (about 63%).
- Recognise that τ is independent of the initial charge or applied voltage and that full charge or discharge is approached asymptotically, not reached in a finite multiple of τ.
- Select and apply the discharging equation \(x = x_0 e^{-t/CR}\) for charge, potential difference, or current.
- Select and apply the charging equation \(x = x_0(1 - e^{-t/CR})\) for charge and potential difference only.
- Recognise that current decays exponentially as \(I = I_0 e^{-t/CR}\) during both charging and discharging processes.
- Ensure the exponent \(-t/CR\) is dimensionless by using consistent units of seconds for both \(t\) and \(CR\).
- Interpret the differential equation ΔQ/Δt = −Q/CR as the rate of change of charge on a discharging capacitor.
- Explain that the negative sign indicates a decreasing charge and that CR is the time constant τ.
- Use a graph of Q against t to find the gradient at a point, which equals ΔQ/Δt, and relate the initial gradient to −Q₀/CR.
- Use a graph of ln Q against t to obtain a straight line of gradient −1/CR and hence determine τ.
- Describe how a spreadsheet can model the equation by iterative steps: Q_new = Q_old + (ΔQ/Δt) × Δt, with ΔQ/Δt calculated from the current Q.
- Explain that reducing the time step Δt improves the accuracy of the numerical model.
- Recognise the shape of an exponential decay graph: a curve that decreases asymptotically towards zero.
- State that the rate of decay is proportional to the current value, leading to the equation x = x₀e^(−t/CR).
- Explain the constant-ratio property: for equal time intervals, the ratio of successive values is constant.
- Calculate the ratio for a given time interval, for example e^(−1) ≈ 0.37 for an interval of one time constant.
- Use the constant-ratio property to identify exponential decay from a table of data or to determine the time constant.
Examiner Tips
- 💡Treat this heading as a map of the topic and check that both sub-statements are covered in your notes.
- 💡Sketch charge, current and potential difference against time for charging and discharging before attempting calculations.
- 💡Link each graph feature to the physical process, such as why current is largest at the instant the switch closes.
- 💡Check that resistance is in ohms and capacitance in farads before multiplying to find τ.
- 💡Use the initial value and the 1/e point to sketch or identify an exponential decay curve.
- 💡Remember that charging and discharging graphs are mirror images in shape but differ in direction and final value.
- 💡State clearly where each meter is connected and what quantity it measures.
- 💡Explain one advantage of a data-logger, such as automatic recording of many readings or smooth graph plotting.
- 💡Suggest a sensible check, such as repeating the experiment or using a longer time constant, to improve reliability.
- 💡Always convert capacitance to farads and resistance to ohms before calculating τ, and state the unit of τ as seconds.
- 💡When asked for the time constant from a graph, identify the time at which the discharging quantity has fallen to about 37% of its initial value, or the charging quantity has risen to about 63% of its final value.
- 💡Remember that τ is a characteristic time, not a total time; avoid saying the capacitor is fully charged or discharged after one, two or five time constants.
- 💡Always check which quantity is being asked for during charging; use the \((1 - e^{-t/CR})\) form for \(Q\) and \(V\), but the \(e^{-t/CR}\) form for \(I\).
- 💡To find \(\tau\) from a graph, plot \(\ln x\) against \(t\) for a decaying quantity; the gradient is \(-1/CR\).
- 💡Remember that after one time constant, a decaying quantity is about 37% of its initial value, and a rising quantity is about 63% of its final value.
- 💡When using a spreadsheet model, always calculate the rate of change using the current value of Q, then update Q by adding (ΔQ/Δt) × Δt.
- 💡To find τ from a graph, plot ln Q against t for a discharging capacitor; the gradient is −1/CR, so τ = −1/gradient.
- 💡Remember that the initial gradient of a Q–t graph for a discharging capacitor is −Q₀/CR, which can be used to estimate τ if Q₀ is known.
- 💡To test for exponential decay, check whether the ratio of successive values at equal time intervals is constant.
- 💡Remember that after one time constant the value is about 37% of the initial value, after two time constants about 14%, and after three about 5%.
- 💡When plotting a graph to determine the time constant, use ln x against t; a straight line confirms exponential decay and its gradient gives −1/CR.
Common Mistakes
- Treating the heading as a stand-alone fact to memorise rather than a signpost to the sub-statements.
- Learning the exponential equations without being able to describe the physical charging and discharging process.
- Confusing the direction of current during charging with the direction during discharging.
- Thinking the current stays constant during charging; it falls because the potential difference across the resistor decreases as the capacitor voltage rises.
- Confusing the time constant with the time to fully charge or discharge; after one time constant the value has changed by about 63%, not completely.
- Using τ = R/C instead of τ = RC, which gives the wrong unit and magnitude.
- Connecting the voltmeter in series with the capacitor instead of in parallel; a voltmeter must be in parallel to measure potential difference.
- Choosing a very small time constant so the capacitor discharges before readings can be taken; a larger resistor or capacitor slows the change.
- Assuming a data-logger removes all error; the sensor range and sampling rate still affect the quality of the results.
- Thinking that τ is the time to fully charge or discharge the capacitor. Correction: τ is the time to reach about 63% of the final value when charging, or to fall to about 37% of the initial value when discharging; the process continues asymptotically.
- Using C in microfarads or R in kilohms without converting to farads and ohms. Correction: convert all values to base SI units before multiplying, for example 470 μF = 470 × 10⁻⁶ F and 10 kΩ = 10 × 10³ Ω.
- Believing that τ depends on the initial voltage or charge. Correction: τ = CR depends only on the capacitance and resistance; changing the initial conditions changes the starting values but not the time constant.
- Confusing the charging and discharging percentages. Correction: when discharging, the value falls to 1/e (about 37%) after one time constant; when charging, it rises to 1 − 1/e (about 63%) after one time constant.
- Assuming current rises during charging; correction: current always decays exponentially in both charging and discharging circuits, following \(I = I_0 e^{-t/CR}\).
- Using the charging equation for a discharging process or vice versa; correction: use \(x = x_0 e^{-t/CR}\) for decreasing quantities and \(x = x_0(1 - e^{-t/CR})\) for increasing quantities.
- Treating \(x_0\) as the final value in the discharging equation; correction: in \(x = x_0 e^{-t/CR}\), \(x_0\) is the initial value at \(t = 0\).
- Ignoring the negative sign in ΔQ/Δt = −Q/CR. Correction: the negative sign shows that charge is decreasing; when modelling, ensure the change in charge is subtracted from the current value.
- Confusing the gradient of a Q–t graph with the time constant. Correction: the gradient at a point is ΔQ/Δt, not τ; the time constant is found from the gradient of ln Q against t, which is −1/CR.
- Using a large time step in a spreadsheet model and expecting accurate results. Correction: the numerical approximation improves as Δt becomes smaller; a large Δt leads to significant error.
- Assuming the differential equation applies to charging as well as discharging. Correction: this form with a negative sign applies to discharging; for charging, the rate of change is positive and the equation is ΔQ/Δt = (Q₀ − Q)/CR.
- Thinking that exponential decay reaches zero in a finite time. Correction: the graph approaches zero asymptotically; it never actually reaches zero in theory.
- Assuming that the ratio of successive values is constant for any time intervals, not just equal intervals. Correction: the constant-ratio property applies only when the time intervals are equal.
- Confusing the constant ratio with a constant difference. Correction: in exponential decay, the difference between successive values decreases, but the ratio remains constant for equal time intervals.
- Believing that the time constant is the time for the quantity to halve. Correction: the half-life is τ ln 2 ≈ 0.693τ, not τ itself.