Welcome to Capacitors!
Welcome to one of the most exciting and scoring topics in Unit A2 2: Fields, Capacitors and Particle Physics (Topic 5.4). Whether you are already confident with electric circuits or find circuit mathematics a bit daunting, do not worry! In these notes, we will break down every single concept into bite-sized, crystal-clear steps to ensure you can master every question CCEA throws at you.
In simple terms, a capacitor is an electrical component designed to store electric charge and electrical potential energy. Think of a capacitor as a tiny, ultra-fast rechargeable energy tank. Unlike chemical batteries, which release energy slowly via chemical reactions, capacitors can charge up and dump their stored energy in fractions of a second.
What you will learn in this chapter:
• What capacitance is and how capacitors are constructed.
• How to calculate the energy stored in an electrostatic field.
• How capacitors behave when connected in series and parallel.
• The mathematics of charging and discharging circuits using the time constant \(\tau = RC\).
• How to use logarithmic plots to extract circuit values.
1. Core Definitions & Physical Principles
What is Inside a Capacitor?
A standard parallel-plate capacitor consists of two parallel conducting plates separated by an insulating material called a dielectric.
When a capacitor is connected across a direct current (\(\text{d.c.}\)) power supply of potential difference \(V\):
• Electrons are pulled away from the plate connected to the positive terminal, leaving it with a net positive charge (\(+Q\)).
• An equal number of electrons are deposited onto the plate connected to the negative terminal, giving it an equal net negative charge (\(-Q\)).
• Current flows temporarily until the potential difference across the capacitor plates equals the supply voltage \(V\). At this point, the capacitor is fully charged, and no further current flows.
Defining Capacitance
Capacitance (\(C\)) is defined as the charge stored per unit potential difference across the plates.
\(C = \frac{Q}{V}\)
Where:
• \(C\) = capacitance in farads (\(\text{F}\))
• \(Q\) = magnitude of charge stored on one plate in coulombs (\(\text{C}\))
• \(V\) = potential difference across the plates in volts (\(\text{V}\))
The Unit: The Farad (\(\text{F}\))
Definition of the Farad: One farad is the capacitance of a conductor/capacitor that stores one coulomb of charge per volt of potential difference across it (\(1\text{ F} = 1\text{ C V}^{-1}\)).
Exam Tip & Common Pitfall: A farad is a very large unit of capacitance. In exam questions, you will almost always encounter submultiples. Always convert these to base units immediately before calculating:
• Microfarads: \(1\text{ }\mu\text{F} = 1 \times 10^{-6}\text{ F}\)
• Nanofarads: \(1\text{ nF} = 1 \times 10^{-9}\text{ F}\)
• Picofarads: \(1\text{ pF} = 1 \times 10^{-12}\text{ F}\)
Key Takeaway: Capacitance is the ratio of charge to voltage (\(C = \frac{Q}{V}\)). One farad equals one coulomb per volt.
2. Energy Stored in a Capacitor
When charging a capacitor, work must be done against the electrostatic repulsive forces of the electrons already sitting on the negative plate. This work done is stored as electrical potential energy in the electrostatic field between the plates.
The Charge vs. Potential Difference (\(Q\text{–}V\)) Graph
If you plot charge \(Q\) on the y-axis against potential difference \(V\) on the x-axis for a capacitor:
• The gradient of the line represents the capacitance (\(C = \frac{\Delta Q}{\Delta V}\)).
• The area under the line represents the work done or energy stored (\(E\)).
Because the graph is a straight line passing through the origin, the area is a triangle with \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\):
\(E = \frac{1}{2}QV\)
Three Equivalent Formulae for Energy Stored
By substituting \(Q = CV\) into the base formula, we get three convenient equations. You can choose whichever fits the data given in your exam question:
\(E = \frac{1}{2}QV = \frac{1}{2}CV^2 = \frac{Q^2}{2C}\)
The 50% Energy Mystery (A Classic Exam Trap!)
Did you know? When a battery of voltage \(V\) pushes a total charge \(Q\) onto a capacitor, the total energy supplied by the battery is \(W = QV\).
However, the energy actually stored in the capacitor is only \(E = \frac{1}{2}QV\).
Where does the other \(50\%\) go? The remaining half of the energy is dissipated as heat in the resistance of the connecting wires and circuit components during the charging process.
Key Takeaway: Energy stored is given by the area under the \(Q\text{–}V\) graph (\(E = \frac{1}{2}QV = \frac{1}{2}CV^2 = \frac{Q^2}{2C}\)). The battery supplies \(QV\), but only half is stored!
3. Capacitors in Combination
Important Rule to Remember: The combination rules for capacitors are the exact opposite of the combination rules for resistors!
Capacitors in Parallel
When capacitors are connected in parallel side-by-side across a supply:
• The potential difference across each capacitor is identical: \(V_T = V_1 = V_2 = \dots\)
• The total charge supplied is the sum of the individual charges: \(Q_T = Q_1 + Q_2 + \dots\)
• Since \(Q = CV\), we have \(C_{\text{total}}V = C_1V + C_2V + \dots\)
Cancelling \(V\) gives the total capacitance formula:
\(C_{\text{total}} = C_1 + C_2 + C_3 + \dots\)
Why this makes sense: Placing plates in parallel effectively increases the total surface area of the plates, meaning they can store more charge at the same voltage.
Capacitors in Series
When capacitors are connected in a single line one after the other:
• The charge stored on each capacitor is identical due to conservation of charge: \(Q_T = Q_1 = Q_2 = \dots\)
• The total potential difference is shared across the capacitors: \(V_T = V_1 + V_2 + \dots\)
• Since \(V = \frac{Q}{C}\), we have \(\frac{Q}{C_{\text{total}}} = \frac{Q}{C_1} + \frac{Q}{C_2} + \dots\)
Cancelling \(Q\) gives the total capacitance formula:
\(\frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots\)
Memory Trick:
• Capacitors in Parallel = Simple Plus (\(C_1 + C_2\))
• Capacitors in Series = Shared reciprocal fractions (\(\frac{1}{C_1} + \frac{1}{C_2}\))
Key Takeaway: Parallel capacitors add directly (\(C_{\text{total}} = C_1 + C_2\)). Series capacitors combine via reciprocals (\(\frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2}\)).
4. Charging and Discharging Through a Resistor
When a capacitor charges or discharges through a resistor \(R\), the transfer of charge is not instantaneous. It follows an exponential curve over time.
The Time Constant (\(\tau\))
The rate at which a capacitor charges or discharges depends on the resistance \(R\) and the capacitance \(C\). We define the time constant (\(\tau\)) as:
\(\tau = RC\)
Where:
• \(\tau\) = time constant in seconds (\(\text{s}\))
• \(R\) = resistance in ohms (\(\Omega\))
• \(C\) = capacitance in farads (\(\text{F}\))
Physical Meaning of the Time Constant:
• During Discharge: \(\tau\) is the time taken for the charge, voltage, or current to fall to \(\frac{1}{e}\) (approximately \(37\%\)) of its initial value.
• During Charging: \(\tau\) is the time taken for the charge or voltage to rise to \(\left(1 - \frac{1}{e}\right)\) (approximately \(63\%\)) of its maximum value.
Discharging a Capacitor (Exponential Decay)
When a charged capacitor is connected across a resistor, it discharges. The charge, voltage, and current all decay exponentially according to the same mathematical form:
\(Q = Q_0 e^{-\frac{t}{RC}}\)
\(V = V_0 e^{-\frac{t}{RC}}\)
\(I = I_0 e^{-\frac{t}{RC}}\)
Where \(Q_0\), \(V_0\), and \(I_0\) represent the initial values at time \(t = 0\), and \(I_0 = \frac{V_0}{R}\).
Charging a Capacitor
When an uncharged capacitor is connected in series with a resistor and a \(\text{d.c.}\) voltage supply \(V_0\):
• Charge and Voltage grow towards their maximum values (\(Q_0\) and \(V_0\)):
\(Q = Q_0\left(1 - e^{-\frac{t}{RC}}\right)\)
\(V = V_0\left(1 - e^{-\frac{t}{RC}}\right)\)
• Charging Current starts at a maximum (\(I_0 = \frac{V_0}{R}\)) at \(t = 0\) and decays exponentially to zero as the capacitor fills up:
\(I = I_0 e^{-\frac{t}{RC}}\)
Examiner Alert: Never draw the charging current graph as increasing! As the capacitor charges, its opposing potential difference builds up, which reduces the net potential difference driving the current. Therefore, current always starts at its peak and decays towards zero during charging.
Logarithmic Analysis (Linearising Discharge Data)
To determine the time constant or capacitance experimentally, we linearise the discharge equation using natural logarithms (\(\ln\)).
Starting with \(V = V_0 e^{-\frac{t}{RC}}\), take the natural log of both sides:
\(\ln V = \ln\left(V_0 e^{-\frac{t}{RC}}\right)\)
\(\ln V = \ln V_0 - \frac{t}{RC}\)
Rearranging to match the standard straight-line equation \(y = mx + c\):
\(\ln V = \left(-\frac{1}{RC}\right)t + \ln V_0\)
If you plot a graph of \(\ln V\) (on the y-axis) against \(t\) (on the x-axis):
• The graph is a straight line with a negative gradient.
• \(\text{Gradient} = -\frac{1}{RC} = -\frac{1}{\tau}\)
• \(\text{y-intercept} = \ln V_0\)
This allows you to calculate \(RC\) directly from \(\text{gradient} = -\frac{1}{RC} \implies RC = -\frac{1}{\text{gradient}}\).
Key Takeaway: Time constant \(\tau = RC\). Discharging follows \(e^{-\frac{t}{RC}}\) (falling to \(37\%\) at \(1\tau\)). Charging voltage follows \(1 - e^{-\frac{t}{RC}}\) (rising to \(63\%\) at \(1\tau\)), while charging current always decays exponentially from \(I_0 = \frac{V_0}{R}\).
5. Top CCEA Exam Pitfalls & How to Avoid Them
1. Stating Incomplete Definitions:
Do not define capacitance as "the amount of charge a capacitor holds". You must define it as the charge stored per unit potential difference (\(C = \frac{Q}{V}\)). Similarly, define the farad as one coulomb per volt.
2. Resistor vs. Capacitor Formula Mix-up:
Always double-check your circuit combinations! Series capacitors use reciprocals (\(\frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2}\)), while parallel capacitors add directly (\(C_{\text{total}} = C_1 + C_2\)).
3. Forgetting Prefix Conversions:
Always check if capacitance is given in \(\mu\text{F}\) (\(10^{-6}\)), \(\text{nF}\) (\(10^{-9}\)), or \(\text{pF}\) (\(10^{-12}\)) before substituting into \(\tau = RC\) or \(E = \frac{1}{2}CV^2\).
4. Energy Formula Confusion:
Remember that energy stored on the plates is \(E = \frac{1}{2}QV\), NOT \(QV\). The total work done by the battery is \(QV\), with \(50\%\) lost as heat.
5. Current Behavior During Charging:
When a capacitor charges, voltage and charge rise exponentially according to \(\left(1 - e^{-\frac{t}{RC}}\right)\), but the current decays exponentially according to \(e^{-\frac{t}{RC}}\).
Quick Reference Summary
• Capacitance: \(C = \frac{Q}{V}\) (Unit: Farad, \(\text{F} = \text{C V}^{-1}\))
• Energy Stored: \(E = \frac{1}{2}QV = \frac{1}{2}CV^2 = \frac{Q^2}{2C}\) (Area under \(Q\text{–}V\) graph)
• Parallel: \(C_{\text{total}} = C_1 + C_2 + C_3 + \dots\)
• Series: \(\frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots\)
• Time Constant: \(\tau = RC\)
• Discharge Equations: \(X = X_0 e^{-\frac{t}{RC}}\) (where \(X\) can be \(Q, V,\) or \(I\))
• Charging Equations: \(V = V_0\left(1 - e^{-\frac{t}{RC}}\right)\), \(Q = Q_0\left(1 - e^{-\frac{t}{RC}}\right)\), \(I = I_0 e^{-\frac{t}{RC}}\)
• Linearised Discharge: \(\ln V = \left(-\frac{1}{RC}\right)t + \ln V_0\)