📚 A-LEVEL PHYSICS (9702) STUDY NOTES: CAPACITANCE (TOPIC 19)
👋 Introduction: Why Capacitance Matters
Welcome to the chapter on Capacitance! This is a core A-Level topic that links electric fields, circuits, and energy storage. Don't worry if the formulas look a bit complex initially; the concepts are highly logical.
Capacitors are essential components in almost all electronic circuits. They are basically tiny energy reservoirs. Think about the massive burst of light from a camera flash, the smooth flow of power in your phone, or the tuning of a radio—all rely heavily on capacitors. By the end of this topic, you'll understand how these devices store charge and release energy instantly!
19.1 Capacitors and Capacitance
1. Defining Capacitance (The 'Size' of the Storage Container)
A capacitor is an electrical component designed to store electrical charge. Physically, it consists of two conducting plates separated by an insulating material called a dielectric (like air, paper, or ceramic).
The key concept here is Capacitance (C).
Definition: Capacitance is defined as the charge stored per unit potential difference (p.d.) across the capacitor. It measures how effective a capacitor is at storing charge for a given voltage.
Analogy: Think of capacitance like the size of a water bucket. A large bucket (high capacitance) can hold a lot of water (charge, Q) before the water pressure (potential difference, V) gets very high.
The Fundamental Equation
The definition gives us the key formula relating charge \(Q\), potential difference \(V\), and capacitance \(C\):
$$C = \frac{Q}{V}$$
- C: Capacitance (Unit: Farad, F)
- Q: Charge stored (Unit: Coulomb, C)
- V: Potential Difference (Unit: Volt, V)
Key Point: One Farad is defined as the capacitance when one Coulomb of charge is stored per one Volt of potential difference (1 F = 1 C V\(^{-1}\)).
Units of Capacitance (Practical Sizes)
The Farad (F) is a very large unit. In practice, capacitors usually have much smaller values:
- microfarad (\(\mu\text{F}\)): \(1 \times 10^{-6} \text{ F}\)
- nanofarad (\(\text{nF}\)): \(1 \times 10^{-9} \text{ F}\)
- picofarad (\(\text{pF}\)): \(1 \times 10^{-12} \text{ F}\)
1. Isolated spherical conductors.
2. Parallel plate capacitors (the most common type you study).
2. Combining Capacitors in Circuits
When capacitors are connected together, their total (resultant) capacitance changes. This is the opposite of how resistors combine!
2.1 Capacitors in Parallel
When capacitors are connected in parallel, they are connected across the same two points.
- Potential Difference (V): This is the same across all components in parallel. \(V_{total} = V_1 = V_2 = V_3\).
- Charge (Q): The total charge stored is the sum of the charges stored on individual capacitors. \(Q_{total} = Q_1 + Q_2 + Q_3\).
Derivation (Required by Syllabus):
1. Start with the conservation of charge: \(Q_{total} = Q_1 + Q_2 + Q_3\)
2. Substitute \(Q = C V\) for each: \(C_{total}V = C_1V_1 + C_2V_2 + C_3V_3\)
3. Since \(V\) is common (constant) in parallel: \(C_{total}V = C_1V + C_2V + C_3V\)
4. Cancel \(V\):
$$C_{total} = C_1 + C_2 + C_3 + \dots$$
Key Takeaway: Connecting capacitors in parallel increases the total effective plate area, thus increasing the total capacitance.
2.2 Capacitors in Series
When capacitors are connected in series, they are connected end-to-end.
- Charge (Q): The charge on each capacitor is the same. This is because the charges must be conserved across the interconnected plates. \(Q_{total} = Q_1 = Q_2 = Q_3\).
- Potential Difference (V): The total p.d. is the sum of the p.d. across each capacitor. \(V_{total} = V_1 + V_2 + V_3\).
Derivation (Required by Syllabus):
1. Start with the conservation of energy (Kirchhoff's Second Law): \(V_{total} = V_1 + V_2 + V_3\)
2. Substitute \(V = Q/C\) for each: \(\frac{Q}{C_{total}} = \frac{Q_1}{C_1} + \frac{Q_2}{C_2} + \frac{Q_3}{C_3}\)
3. Since \(Q\) is common (constant) in series: \(\frac{Q}{C_{total}} = \frac{Q}{C_1} + \frac{Q}{C_2} + \frac{Q}{C_3}\)
4. Cancel \(Q\):
$$\frac{1}{C_{total}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots$$
Key Takeaway: Connecting capacitors in series decreases the total capacitance. The total capacitance is always less than the smallest individual capacitance.
Capacitors behave the opposite way to Resistors:
- Resistors in Series: Add (R = R1 + R2)
- Capacitors in Series: Reciprocal (1/C = 1/C1 + 1/C2)
- Resistors in Parallel: Reciprocal (1/R = 1/R1 + 1/R2)
- Capacitors in Parallel: Add (C = C1 + C2)
19.2 Energy Stored in a Capacitor
1. How Energy is Stored (The Q-V Graph)
When charging a capacitor, work must be done to move charge (electrons) from one plate to the other against the electrostatic repulsive force. This work done is stored as electric potential energy in the electric field between the plates.
To find the total work done (\(W\)) or energy stored (\(E\)), we look at the graph of potential difference (\(V\)) against charge (\(Q\)).
- Initially, \(V = 0\), so little work is needed to move the first bit of charge.
- As charge builds up, \(V\) increases (since \(V = Q/C\)), so more work is required for subsequent charge movement.
Because \(V\) is proportional to \(Q\), the graph of \(V\) vs \(Q\) is a straight line passing through the origin.
Syllabus Requirement: The electric potential energy stored is determined by the area under the potential-charge graph (V vs Q).
Since the area is a triangle (Area = 1/2 × base × height):
$$W = \frac{1}{2} \times Q \times V$$
2. Energy Storage Formulas
Using the fundamental relationship \(Q = CV\), we can derive two other useful forms for the stored energy \(W\):
Formula 1 (Fundamental):
$$W = \frac{1}{2} Q V$$
Formula 2 (Substituting \(Q=CV\)):
$$W = \frac{1}{2} C V^2$$
Formula 3 (Substituting \(V=Q/C\)):
$$W = \frac{Q^2}{2 C}$$
You must be able to recall and use all three forms. Often, \(W = 1/2 CV^2\) is the most practical formula, as \(C\) is constant and \(V\) is easy to measure.
Do not confuse \(W = 1/2 QV\) with the general power formula \(P=VI\) or energy \(E=VIt\). Those formulas apply when V is constant, but V across a charging capacitor is not constant! That's why we need the factor of \(1/2\).
19.3 Discharging a Capacitor (RC Circuits)
When a charged capacitor is connected across a resistor (\(R\)), the charge stored begins to flow through the resistor, converting the stored electrical energy into heat. This is known as an RC circuit.
1. Exponential Decay
The key characteristic of capacitor discharge is that the rate of discharge is proportional to the amount of charge remaining. This leads to an exponential decay for the charge (\(Q\)), potential difference (\(V\)), and current (\(I\)) over time.
The mathematical form of this decay is given by:
$$x = x_0 e^{-t/RC}$$
Where:
- \(x\) is the quantity (Charge \(Q\), Voltage \(V\), or Current \(I\)) at time \(t\).
- \(x_0\) is the initial quantity (at \(t=0\)).
- \(e\) is the base of the natural logarithm (\(\approx 2.718\)).
- \(RC\) is the Time Constant.
Analysing Discharge Graphs
When discharging, the graphs of \(V\), \(Q\), and \(I\) against time all follow the same shape:
They start at their maximum value (\(x_0\)) and drop sharply at first, before the rate of decrease slows down as they approach zero. This is a classic exponential decay curve.
2. The Time Constant (\(\tau\))
The speed at which a capacitor discharges depends on the values of the resistance \(R\) and the capacitance \(C\).
Definition: The time constant, \(\tau\), is defined as the product of the resistance \(R\) and the capacitance \(C\).
$$\tau = RC$$
The time constant has the unit of seconds (s) (Check this using base units: \(\text{Ohm} \times \text{Farad} = (\text{V}/\text{A}) \times (\text{C}/\text{V}) = \text{C}/\text{A} = \text{s}\)).
Physical Meaning of \(\tau\)
The time constant \(\tau\) is the time required for the charge (\(Q\)), potential difference (\(V\)), or current (\(I\)) to fall to \(1/e\) (approximately 37%) of its initial value.
Example: If the initial voltage is \(10.0 \text{ V}\) and the time constant is \(2.0 \text{ s}\), after \(2.0 \text{ s}\) the voltage will drop to \(10.0 \times e^{-1} \approx 3.7 \text{ V}\).
Factors Affecting Discharge Speed
A larger time constant (\(\tau\)) means the capacitor discharges more slowly.
- Increasing R: Makes discharge slower (higher resistance restricts current flow).
- Increasing C: Makes discharge slower (higher capacitance means more charge has to flow out).
In power supplies, AC is converted to DC using rectification. However, the DC output is often wavy (rippled). A capacitor connected in parallel with the load resistor is used for smoothing. During the peaks of the AC cycle, the capacitor charges up. When the voltage drops, the capacitor discharges slowly (due to the large load resistance \(R\)), filling in the gaps and making the output voltage much steadier. The time constant \(RC\) must be much larger than the period of the AC supply for effective smoothing.
3. Determining \(\tau\) from Graphs
In examination questions, you may be asked to determine the time constant from a discharge graph (V-t, Q-t, or I-t).
Step-by-Step Method:
1. Identify Initial Value (\(x_0\)): Find the value of \(V\), \(Q\), or \(I\) at \(t=0\).
2. Calculate 37% Value: Calculate the value \(x = 0.37 \times x_0\).
3. Read Time (\(\tau\)): Locate this value (\(x\)) on the y-axis, trace horizontally to the curve, and then trace vertically down to the time axis to read the value of \(\tau\).
Alternatively, you can use the gradient method: The initial gradient of the discharge curve is \(-x_0 / \tau\). If you draw a tangent to the curve at \(t=0\), the time taken for this tangent to reach the time axis (where \(x=0\)) is exactly equal to the time constant \(\tau\).
Key Takeaways for Capacitance
This chapter revolves around four main formulas and their applications:
- Definition: \(C = Q/V\)
- Energy: \(W = 1/2 Q V = 1/2 C V^2\)
- Series/Parallel Rules: Remember the rules are swapped compared to resistors.
- Discharge/Time Constant: \(\tau = RC\) and \(x = x_0 e^{-t/\tau}\)