Welcome to the World of Capacitors!
Hello future physicist! This chapter moves us beyond simple flowing current (DC circuits) and introduces a fundamental component used throughout modern electronics: the capacitor.
Think of a capacitor as an ultra-fast, temporary energy storage device. While a battery stores energy chemically and releases it slowly, a capacitor stores energy electrically (as separated charge) and can release it almost instantly. This ability makes them essential for things like camera flashes, smoothing power signals, and memory chips.
Don't worry if the math looks intimidating at first—we'll break down the concepts, especially the scary-sounding "exponential decay," into simple, manageable steps!
3.8.4 The Basics: Defining Capacitance (C)
What is a Capacitor?
A simple capacitor typically consists of two conducting plates separated by a small gap of insulating material called a dielectric.
- When connected to a voltage source (like a battery), one plate accumulates positive charge (+Q) and the other accumulates an equal amount of negative charge (-Q).
- The capacitor itself stores a total charge of Q (the magnitude of charge on either plate).
Definition of Capacitance
Capacitance (C) is a measure of a capacitor’s ability to store charge. Specifically, it tells us how much charge (Q) can be stored per unit potential difference (V) across its plates.
The definition is given by the formula:
$$C = \frac{Q}{V}$$
- Q: Charge stored (in Coulombs, C)
- V: Potential difference / voltage across the plates (in Volts, V)
- C: Capacitance (in Farads, F)
Did you know? The unit of capacitance, the Farad (F), is defined as one Coulomb per Volt ($$\text{1 F} = 1 \text{ C V}^{-1}$$). A capacitor with a capacitance of 1 Farad is actually enormous in practice! Most components you use will be measured in microfarads (\(\mu\text{F}\), $$10^{-6}\text{ F}$$) or picofarads (\(\text{pF}\), $$10^{-12}\text{ F}$$).
Key Takeaway
Capacitance is the ratio of charge stored to the voltage applied. The higher the capacitance, the more charge it can hold for the same applied voltage.
3.8.4 Factors Affecting Capacitance
For a specific type of capacitor—the parallel plate capacitor—we can calculate the capacitance based purely on its physical dimensions and the material between the plates.
$$C = \frac{A\epsilon_0 \epsilon_r}{d}$$
- A: Area of overlap between the plates (m\(^2\)). More area means more room for charge separation, so C increases.
- d: Separation distance between the plates (m). A smaller distance means the charges attract each other more strongly across the gap, helping to store more charge, so C increases.
- \(\epsilon_0\): The permittivity of free space (a constant).
- \(\epsilon_r\): The relative permittivity (or dielectric constant).
Quick Review: The key to increasing C is to have a Large Area (A), a Small Separation (d), and a good Dielectric (\(\epsilon_r\)).
3.8.4 Dielectrics: Why Insulation Matters
Relative Permittivity (\(\epsilon_r\))
The material placed between the plates is called the dielectric (usually an insulator like paper, plastic, or air). The factor \(\epsilon_r\) tells us how much better the capacitance is when that material is present compared to a vacuum.
- For a vacuum, \(\epsilon_r = 1\).
- For air, \(\epsilon_r\) is very close to 1.
- For other insulating materials, \(\epsilon_r > 1\).
The Action of a Dielectric
How does putting an insulator in the gap actually increase the capacitor's ability to store charge?
The secret lies in polar molecules:
- When a voltage is applied, an electric field (E-field) is set up between the plates (pointing from + to -).
- If the dielectric material contains simple polar molecules (molecules that have a slightly positive end and a slightly negative end, like water), these molecules will rotate to align themselves with the external E-field.
- This alignment creates a small, internal electric field within the dielectric that opposes the main field from the plates.
- This opposition means the net electric field between the plates is reduced.
- Since the voltage \(V = Ed\) (in a uniform field), a reduced E means the voltage V across the plates is lowered for the same stored charge Q.
- Since \(C = Q/V\), lowering V (while Q stays constant) means the Capacitance (C) increases.
Key Takeaway
Dielectrics don't conduct charge, but they allow the plates to hold more charge for a given voltage by weakening the electric field between them.
3.8.4 Energy Stored in a Capacitor
When charging a capacitor, work must be done to move charge from one plate to the other against the existing electric field. This work done is stored as electric potential energy (E).
Graphical Interpretation
If you plot the Charge (Q) on the plates against the Potential Difference (V) across the plates, you get a straight line passing through the origin (since \(Q = CV\)).
- The area under the Q-V graph represents the total work done to charge the capacitor, and thus the Energy Stored (E).
Since the area is a triangle (\(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\)):
$$E = \frac{1}{2}QV$$
The Three Energy Formulas
Since \(Q = CV\), we can substitute this relationship to get two other useful expressions for energy stored:
- Using Q and V: $$E = \frac{1}{2}QV$$
- Substituting Q = CV: $$E = \frac{1}{2}CV^2$$
- Substituting \(V = Q/C\): $$E = \frac{1}{2}\frac{Q^2}{C}$$
Memory Aid: You usually use the formula that contains the variables you know or need to find. The formula \(E = \frac{1}{2}CV^2\) is often the most practical in circuit problems as C is constant and V is easy to measure.
Key Takeaway
Energy storage is proportional to the square of the voltage or the charge. Double the voltage, and you quadruple the stored energy!
3.9 Exponential Change: Capacitors in RC Circuits (A-level only)
When a capacitor is connected in series with a resistor (R) and a DC power supply, the process of charging and discharging does not happen instantaneously; it happens exponentially. This combination is known as an RC circuit.
3.9.1 The Time Constant (RC)
Defining the Time Constant (\(\tau\))
The speed at which a capacitor charges or discharges depends entirely on the values of the resistance (R) and the capacitance (C) in the circuit. Their product is the Time Constant (\(\tau\)).
$$\tau = RC$$
- R: Resistance (Ohms, \(\Omega\))
- C: Capacitance (Farads, F)
- \(\tau\): Time constant (Seconds, s)
Analogy: If the capacitor is a water tank, the voltage is the water level, and the charge is the volume of water. The resistor is like a narrow pipe restricting the flow. A high R or a high C (a huge tank or a very narrow pipe) will make the charging/discharging process much slower, leading to a larger time constant.
Physical Significance of \(\tau\)
- In Discharging: \(\tau\) is the time taken for the charge (Q) and the voltage (V) to fall to $$1/e$$ (approximately 37%) of their initial value.
- In Charging: \(\tau\) is the time taken for the charge (Q) and the voltage (V) to rise to $$(1 - 1/e)$$ (approximately 63%) of their maximum (final) value.
Time to Halve ($$T_{1/2}$$)
It is often useful to calculate the "half-life" (time to fall to half the initial value), which is related to \(\tau\):
$$T_{\frac{1}{2}} = \ln(2)RC \quad \text{or} \quad T_{\frac{1}{2}} \approx 0.693 RC$$
Key Takeaway
The time constant (\(\tau\)) dictates the pace of the exponential process. A larger \(\tau\) means a slower charge/discharge.
3.9.1 Graphical Representation and Interpretation
The graphs of Q, V, and I against time (t) are essential. They are all exponential curves.
1. Discharging (The capacitor is losing charge)
The capacitor starts fully charged ($$Q_0$$ and $$V_0$$) and the current immediately begins to flow ($$I_0$$).
- Charge (Q) vs. t and Voltage (V) vs. t:
Both Q and V decay exponentially towards zero. The rate of decay is fastest at the start when V is highest (and thus the current I is highest).
- Current (I) vs. t:
The current also decays exponentially towards zero. This makes sense: as V drops, the driving potential for current drops, so I drops.
Interpretation of Gradients and Areas in Discharging Graphs:
- Gradient of the Q-t graph: \( \frac{\Delta Q}{\Delta t} \) is the rate of flow of charge, which equals the current (I). Since the gradient gets less steep over time, the current is decreasing.
- Area under the I-t graph: $$ \int I dt $$ represents the total charge (Q) that flowed off the capacitor.
2. Charging (The capacitor is gaining charge)
The capacitor starts empty (Q=0, V=0) and is connected to a supply voltage ($$V_S$$).
- Charge (Q) vs. t and Voltage (V) vs. t:
Both Q and V increase exponentially towards their maximum final values ($$Q_0$$ and $$V_S$$). The rate of increase slows down as the capacitor approaches full charge (because the capacitor's voltage starts to oppose the supply voltage).
- Current (I) vs. t:
The current starts at a maximum value ($$I_0 = V_S/R$$) and decays exponentially to zero. When the capacitor is fully charged, its voltage equals the supply voltage, meaning no current flows.
Common Mistake to Avoid: Remember, I (current) always decays exponentially in both charging and discharging circuits!
3.9.1 The Quantitative Treatment (The Maths!)
We use exponential equations involving the natural logarithm base \(e\) (where \(e \approx 2.718\)) to describe these processes precisely.
1. Capacitor Discharge Equations
These equations describe how Q, V, and I drop from their initial peak values ($$Q_0, V_0, I_0$$) towards zero.
$$Q = Q_0 e^{-\frac{t}{RC}}$$ $$V = V_0 e^{-\frac{t}{RC}}$$ $$I = I_0 e^{-\frac{t}{RC}}$$
Where \(Q_0, V_0, I_0\) are the initial values at time \(t=0\).
2. Capacitor Charge Equations
These equations describe how Q and V rise towards their maximum values ($$Q_0, V_S$$) and how I drops from its maximum value ($$I_0$$).
- Charge and Voltage (rising):
$$Q = Q_0 \left(1 - e^{-\frac{t}{RC}}\right)$$ $$V = V_S \left(1 - e^{-\frac{t}{RC}}\right)$$
The term $$(1 - e^{-t/RC})$$ means the value approaches the maximum $$(Q_0 \text{ or } V_S)$$ but never quite reaches it theoretically.
- Current (falling):
The current decays exactly as it does during discharge, starting from its maximum value ($$I_0 = V_S/R$$).
$$I = I_0 e^{-\frac{t}{RC}}$$
Required Practical 6 Connection
You can use these exponential discharge equations to determine the time constant, RC, experimentally. If you take the natural logarithm of the voltage discharge equation:
$$\ln(V) = \ln(V_0) - \frac{t}{RC}$$
This is in the form of a straight line, $$y = c + mx$$:
- Plot \(\ln(V)\) (y-axis) against \(t\) (x-axis).
- The y-intercept will be \(\ln(V_0)\).
- The gradient (m) will be equal to $$- \frac{1}{RC}$$.
- By calculating the gradient, you can easily find the time constant $$(RC = -\frac{1}{\text{gradient}})$$. This is called log-linear plotting.
Chapter Summary: Key Takeaways
- Capacitance Definition: $$C = Q/V$$, measured in Farads (F).
- Parallel Plate C: $$C = \frac{A\epsilon_0 \epsilon_r}{d}$$. C increases with area (A) and decreases with separation (d).
- Energy Stored: $$E = \frac{1}{2}QV = \frac{1}{2}CV^2$$.
- Dielectrics: Polar molecules rotate in the E-field, reducing the net V for the same Q, thus increasing C.
- Time Constant: $$\tau = RC$$. Controls the speed of charge/discharge.
- Discharge Maths: $$Q, V, \text{ and } I$$ all decay exponentially: $$X = X_0 e^{-t/RC}$$.
- Charging Maths: $$Q \text{ and } V$$ rise: $$X = X_0(1 - e^{-t/RC})$$. $$I$$ still decays.