Study Notes: Capacitor Charge and Discharge (Exponential Change 9630)
Hello future physicist! This chapter dives into one of the most exciting areas of electricity: how capacitors store and release energy over time. This process is not linear; it follows an
exponential pattern, a concept vital not just for capacitors, but also for radioactive decay and damping in oscillations. Mastering this section means mastering exponential change!
1. The RC Circuit and the Time Constant
When a capacitor (\(C\)) is connected in series with a resistor (\(R\)) and a power supply, it forms an RC circuit. The resistor controls how quickly the capacitor can charge or discharge. This speed is defined by a single, crucial value: the Time Constant.
Definition of Capacitance and The Time Constant
The capacitance (\(C\)) is defined as the charge stored per unit potential difference:
$$C = \frac{Q}{V}$$
The Time Constant (\(\tau\)) is the characteristic time scale for the exponential process.
$$\tau = RC$$
- R is the resistance (in \(\Omega\)).
- C is the capacitance (in Farads, F).
Did you know? Multiplying Ohms by Farads gives you seconds! The unit for the time constant (\(RC\)) is always seconds (s). This is a great way to check your units in calculations.
Physical Meaning of the Time Constant (\(\tau\))
The time constant (\(\tau\)) is defined as:
- The time taken for a discharging capacitor's charge, voltage, or current to fall to \(1/e\) (approximately 36.8% or 37%) of its initial value.
- The time taken for a charging capacitor's charge or voltage to rise to \((1 - 1/e)\) (approximately 63.2% or 63%) of its final, maximum value.
Analogy: The Leaky Bucket
Imagine a bucket (the capacitor) being filled or emptied through a small pipe (the resistor).
A large \(C\) (big bucket) means more charge must move, slowing the process.
A large \(R\) (narrow pipe) means the current (flow rate) is restricted, also slowing the process.
Therefore, a large Time Constant (\(\tau = RC\)) means the capacitor takes a long time to charge or discharge.
2. Discharging a Capacitor (Decay)
Discharging occurs when a fully charged capacitor is disconnected from the power supply and connected across a resistor. The stored energy drives a current through the resistor.
The Exponential Decay Equations
The charge, voltage, and current all decrease exponentially towards zero.
1. Charge (Q)
$$Q = Q_0 e^{-t/RC}$$
2. Voltage (V)
$$V = V_0 e^{-t/RC}$$
3. Current (I)
$$I = I_0 e^{-t/RC}$$
(Where \(Q_0\), \(V_0\), and \(I_0\) are the initial values at \(t=0\).)
Remembering the discharge formula: All quantities are decreasing, so they follow the pure decay form: Original value multiplied by the exponential decay factor.
Graphical Interpretation of Discharge
- Q-t and V-t Graphs: Start at maximum (\(Q_0\) or \(V_0\)) and curve down, approaching zero asymptotically (never truly reaching zero mathematically, but zero for all practical purposes).
- I-t Graph: Starts at maximum current (\(I_0 = V_0/R\)) and curves down to zero.
Interpretation of the Gradient
The rate of change of charge is the current: \(I = \Delta Q / \Delta t\). On the Q-t graph, the steepness (gradient) represents the current.
- At \(t=0\), the gradient is steepest, meaning the current is maximum (\(I_0\)).
- As \(t\) increases, the gradient becomes less steep, meaning the current drops.
- The gradient is proportional to the charge (and voltage) remaining, which is the defining feature of exponential decay.
3. Charging a Capacitor (Build-up)
Charging occurs when a capacitor is connected in series with a resistor and a DC voltage source (\(V_{supply}\)).
The Exponential Growth Equations
While charge and voltage rise, the current falls (because the capacitor's increasing voltage opposes the supply voltage).
1. Maximum Charge (\(Q_0\))
This is the final charge the capacitor reaches if fully charged. \(Q_0 = C V_{supply}\).
2. Charge (Q)
The charge grows towards the maximum charge \(Q_0\):
$$Q = Q_0 (1 - e^{-t/RC})$$
3. Voltage (V)
The voltage across the capacitor grows towards the supply voltage \(V_{supply}\):
$$V = V_{supply} (1 - e^{-t/RC})$$
4. Current (I)
The current decays from its initial maximum value \(I_0 = V_{supply}/R\):
$$I = I_0 e^{-t/RC}$$
Understanding the \(1-e\) Formula: The term \(e^{-t/RC}\) represents the fraction of the final value that is *missing*. Subtracting this from 1 gives the fraction that has been gained.
Graphical Interpretation of Charging
- Q-t and V-t Graphs: Start at zero and curve upwards, approaching the maximum value (\(Q_0\) or \(V_{supply}\)) asymptotically.
- I-t Graph: This looks exactly like the discharge current graph! It starts at \(I_0\) and decays exponentially to zero.
Interpretation of the Area
Since current is the rate of flow of charge, the total charge stored (or removed) is given by the area under the I-t graph (Area = \(I \times t\), which is charge).
For charging, the total shaded area under the decaying current curve represents the total charge \(Q_0\) stored by the capacitor.
In both charging and discharging, the current (I) is always described by an exponential decay formula: \(I = I_0 e^{-t/RC}\).
4. Energy Stored in a Capacitor
The energy stored (\(E\)) in a capacitor is the work done moving charge against the potential difference. Energy stored is also lost exponentially during discharge.
Energy stored can be found from the area under a Charge (Q) versus Potential Difference (V) graph. This area is always triangular.
$$E = \frac{1}{2} Q V$$
Using the definition \(Q=CV\), the energy formulas are:
$$E = \frac{1}{2} C V^2$$
$$E = \frac{Q^2}{2C}$$
The syllabus requires the interpretation of the area under the Q-V graph.
5. Graphical Determination of the Time Constant (Required Practical 6)
One of the most precise ways to determine the time constant (\(RC\)) experimentally is by plotting the data in a way that produces a straight line (log-linear plotting).
Linearising the Discharge Equation
We start with the decay equation for voltage (or charge/current):
$$V = V_0 e^{-t/RC}$$
To convert this into the straight-line form \(y = mx + c\), we take the natural logarithm (\(\ln\)) of both sides:
$$\ln(V) = \ln(V_0) + \ln(e^{-t/RC})$$
$$\ln(V) = \ln(V_0) - \frac{t}{RC}$$
Rearranging this to match \(y = mx + c\):
$$\ln(V) = \left( - \frac{1}{RC} \right) t + \ln(V_0)$$
- y-axis: \(\ln(V)\)
- x-axis: \(t\) (time)
- Gradient (m): \(m = - \frac{1}{RC}\)
- y-intercept (c): \(c = \ln(V_0)\)
Practical Step-by-Step:
- Measure the voltage (\(V\)) across the capacitor or resistor at regular time intervals (\(t\)) during discharge.
- Calculate \(\ln(V)\) for every reading.
- Plot a graph of \(\ln(V)\) (y-axis) against \(t\) (x-axis). This should yield a straight line with a negative gradient.
- Calculate the gradient \(m\) of the straight line (including error bars if required).
- The Time Constant \(\tau\) is found from the gradient:
$$\tau = RC = - \frac{1}{\text{Gradient}}$$
Half-Life of Capacitors
Just like radioactive decay, we can define a half-life (\(T_{1/2}\)) for a capacitor: the time taken for the voltage (or charge) to fall to half its initial value.
The relationship between half-life and the time constant is:
$$T_{1/2} = \ln 2 \times RC$$
Since \(\ln 2 \approx 0.693\):
$$T_{1/2} \approx 0.693 \times RC$$
Plotting \(\ln(V)\) vs. \(t\) turns the exponential curve into a straight line. The gradient of this line is \(-1/RC\). This is the key mathematical trick for analyzing exponential change.