Welcome to Chapter 19: Capacitors and Capacitance!
Hello future physicist! This chapter introduces you to one of the most fundamental components in modern electronics: the capacitor. Don't worry if electricity seems tricky—we’ll break this down using simple analogies.
Capacitors are essential devices used everywhere, from smoothing ripples in power supplies (like in your phone charger) to storing energy for camera flashes and tuning radio circuits. Learning about them connects your AS Level knowledge of charge and potential difference to powerful A Level applications!
19.1 Capacitors and Capacitance: The Basics
What is a Capacitor? (The Charge Reservoir)
A capacitor is essentially a device designed to store electric charge and electrical potential energy.
It typically consists of two parallel conducting plates separated by an insulating material called a dielectric (like air, paper, or plastic).
- When connected to a battery, charge moves: one plate accumulates positive charge (+Q), and the other accumulates an equal amount of negative charge (-Q).
- The potential difference (V) builds up across the plates as charge accumulates.
Analogy: The Water Tank
Imagine a capacitor like a water storage tank.
- The charge (Q) stored in the capacitor is like the volume of water in the tank.
- The potential difference (V) across the plates is like the water pressure (or height) in the tank.
- A good capacitor (high capacitance) is like a wide tank – it can hold lots of water (Q) without the pressure (V) getting too high!
Defining Capacitance (\(C\))
Capacitance (\(C\)) is a measure of a capacitor's ability to store charge. It is defined as the ratio of the charge stored on the plates (\(Q\)) to the potential difference (\(V\)) across them.
Key Definition:
$$C = \frac{Q}{V}$$
Where:
- \(C\) is the Capacitance (the “capacity” to hold charge).
- \(Q\) is the magnitude of charge stored on one plate (in Coulombs, C).
- \(V\) is the potential difference across the plates (in Volts, V).
The SI unit for capacitance is the Farad (F).
1 Farad: A capacitor has a capacitance of 1 Farad if 1 Coulomb of charge causes a potential difference of 1 Volt across its plates.
Did you know? The Farad is a very large unit. Most capacitors used in electronics are measured in microfarads (\(\mu \text{F}\), \(10^{-6}\text{F}\)), nanofarads (\(\text{nF}\), \(10^{-9}\text{F}\)), or picofarads (\(\text{pF}\), \(10^{-12}\text{F}\)).
Key Takeaway for 19.1 (Definition): Capacitance is simply the ratio of charge stored per volt applied (\(C = Q/V\)). Higher C means more charge stored for the same voltage.
19.1 Combining Capacitors in Circuits
Just like resistors, capacitors can be connected in series or parallel, changing the overall equivalent capacitance of the circuit.
1. Capacitors in Parallel
When capacitors are connected in parallel:
- The potential difference (V) across each capacitor is the same (equal to the source voltage).
- The total charge (\(Q\)) stored is the sum of the charges on each capacitor: $$Q_{total} = Q_1 + Q_2 + Q_3 + ...$$
Derivation (Required by Syllabus)
We use the basic equation \(Q = CV\).
1. Replace \(Q_{total}\) with \(C_{parallel} V\) and individual charges with \(C_i V\): $$C_{parallel} V = C_1 V + C_2 V + C_3 V$$
2. Since \(V\) is the same for all components, we can cancel it out:
Formula for Parallel Capacitance:
$$C_{parallel} = C_1 + C_2 + C_3 + ...$$
Think of it simply: Connecting capacitors in parallel increases the effective area of the plates, so the total capacitance increases.
2. Capacitors in Series
When capacitors are connected in series:
- The charge (Q) stored on each capacitor is the same. (Charge flows out of the battery onto the first plate, inducing an equal and opposite charge on the second plate, and so on).
- The source voltage (\(V\)) is shared across the components: $$V_{total} = V_1 + V_2 + V_3 + ...$$
Derivation (Required by Syllabus)
We rearrange the basic equation to \(V = Q/C\).
1. Replace \(V_{total}\) with \(Q/C_{series}\) and individual voltages with \(Q/C_i\): $$\frac{Q}{C_{series}} = \frac{Q}{C_1} + \frac{Q}{C_2} + \frac{Q}{C_3}$$
2. Since \(Q\) is the same for all components, we can cancel it out:
Formula for Series Capacitance:
$$\frac{1}{C_{series}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + ...$$
Quick Review & Memory Aid
This is the opposite of resistor rules!
- For Parallel: Capacitance ADD (\(C_P = C_1 + C_2\)).
- For Series: Capacitance RECIPROCALS ADD (\(1/C_S = 1/C_1 + 1/C_2\)).
Key Takeaway for 19.1 (Combinations): Capacitors in parallel increase total capacitance; capacitors in series decrease total capacitance. Remember that the formulas are flipped compared to resistors.
19.2 Energy Stored in a Capacitor
To charge a capacitor, you must move charge from one plate (where it is already crowded with like charge) to the other (where there is opposing charge). This requires doing work (W), and this work done is stored as electrical potential energy (\(W\)).
Determining Energy from the Potential-Charge Graph
When charging a capacitor, the potential difference (\(V\)) is constantly increasing (since \(V=Q/C\)). We cannot simply use \(W = QV\). Instead, we consider the average voltage.
- We plot the potential difference (\(V\)) on the y-axis against the charge (\(Q\)) on the x-axis.
- The graph is a straight line passing through the origin (since \(V \propto Q\)).
- The gradient of this graph is \(V/Q\), which is equal to \(1/C\).
- The area under the potential-charge graph represents the total work done (energy stored).
Since the graph forms a triangle up to charge \(Q\) and potential \(V\), the area is: $$W = \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ $$W = \frac{1}{2}QV$$
Energy Stored Formulae (Recalling and Using)
The syllabus requires you to recall and use the following forms, which are all derived from substituting \(C = Q/V\) into \(W = \frac{1}{2}QV\):
1. Basic Form (most often used): $$W = \frac{1}{2}QV$$
2. In terms of C and V: (Substitute \(Q = CV\) into the basic form) $$W = \frac{1}{2}(CV)V$$ $$W = \frac{1}{2}CV^2$$
3. In terms of Q and C: (Substitute \(V = Q/C\) into the basic form) $$W = \frac{1}{2}Q \left(\frac{Q}{C}\right)$$ $$W = \frac{1}{2}\frac{Q^2}{C}$$
Common Mistake Alert! Do not confuse the capacitor energy formula with the general electrical energy formula \(E = QV\). Because the voltage increases from zero to V while charging, the average voltage is only \(\frac{1}{2}V\), hence the factor of \(\frac{1}{2}\).
Key Takeaway for 19.2: The energy stored (W) is calculated by finding the area under the V-Q graph, leading to the master formula \(W = \frac{1}{2}CV^2\).
19.3 Discharging a Capacitor: RC Circuits
When a fully charged capacitor is connected across a resistor (\(R\)), the charge stored begins to flow out, creating a current. This process is called discharging.
The rate at which the capacitor discharges depends on the resistance and the capacitance, forming an RC circuit.
1. Exponential Decay (Analyzing Graphs)
As the capacitor discharges, the charge (\(Q\)) decreases, which means the potential difference (\(V\)) across it decreases. Since \(V=IR\), the current (\(I\)) flowing through the resistor also decreases.
The discharge is exponential, meaning it decreases quickly at first and then slows down.
The graphs showing the variation of \(Q\), \(V\), and \(I\) with time (\(t\)) all follow the same characteristic exponential decay curve:
- The curves start high (at their initial maximum values \(Q_0\), \(V_0\), or \(I_0\)).
- They drop sharply at \(t=0\).
- They flatten out, approaching zero asymptotically (never quite reaching it theoretically).
2. The Time Constant (\(\tau\))
How fast does it decay? This is governed by the Time Constant, \(\tau\).
Definition: The time constant (\(\tau\)) is the time taken for the charge, current, or potential difference of a discharging capacitor to fall to \(1/e\) (about 37%) of its initial maximum value.
The formula for the time constant is incredibly simple:
$$\tau = RC$$
Where \(R\) is the resistance (in \(\Omega\)) and \(C\) is the capacitance (in F). Note that the unit for \(RC\) is seconds (s).
Physical Meaning:
- A large \(R\) (high resistance) means the current is low, so the capacitor discharges slowly (large \(\tau\)).
- A large \(C\) (high capacity) means there is more charge to move, so it takes longer to discharge (large \(\tau\)).
3. The Exponential Equations (Using \(x = x_0 e^{-t/RC}\))
The mathematical relationship describing exponential decay is vital. You must be able to use the equation of the form:
$$x = x_0 e^{-t/RC}$$
Where:
- \(x\) is the value of the quantity (Q, V, or I) at time \(t\).
- \(x_0\) is the initial value of the quantity (Q, V, or I) at time \(t=0\).
- \(e\) is the base of the natural logarithm (\(e \approx 2.718\)).
- \(RC\) is the time constant (\(\tau\)).
This single equation applies to all three variables during discharge:
Charge: \(Q = Q_0 e^{-t/RC}\)
Potential Difference: \(V = V_0 e^{-t/RC}\)
Current: \(I = I_0 e^{-t/RC}\)
Tip for calculations: If a question asks for the charge remaining after one time constant (i.e., when \(t=RC\)): $$Q = Q_0 e^{-RC/RC} = Q_0 e^{-1}$$ $$Q \approx 0.368 Q_0 \quad (36.8\% \text{ of the original charge)}$$
Key Takeaway for 19.3: Discharge is exponential, governed by the time constant \(\tau = RC\). Remember the exponential decay formula \(x = x_0 e^{-t/RC}\) applies equally to charge, voltage, and current.