# Capacitor Charge and Discharge ## Charging a Capacitor - Capacitors are charged by connecting them to a **power supply** (e.g. a battery) - During charging, electrons are **pulled** from the plate connected to the **positive terminal** of the power supply - This plate becomes **positively charged** - Electrons travel around the circuit and are **pushed** onto the plate connected to the **negative terminal** - This plate becomes **negatively charged** - As negative charge builds up on one plate, **electrostatic repulsion** makes it harder to push more electrons onto that plate - Fewer electrons flow as time progresses - Charging stops when no more electrons can be pushed onto the negative plate - The capacitor has reached its **maximum charge**

A parallel plate capacitor consists of two conductive plates with equal but opposite charges building up on each plate

### Charging Current - At the start of charging, the **current is large** - Current gradually **decreases to zero** as electrons stop flowing through the circuit - The current decreases **exponentially** - The rate at which current decreases is proportional to the amount of current remaining - Similar to how a hot cup of tea cools faster when it's very hot compared to when it's lukewarm ### Charging Voltage and Charge - As equal but opposite charges build up on each plate, the **potential difference** between the plates slowly increases - Eventually reaches the same p.d. as the power supply - Similarly, the **charge** on the plates slowly increases - Eventually reaches its maximum value, defined by the capacitance of the capacitor ### Charging Graphs | Quantity | Graph Shape | Initial Value | Final Value | |----------|-------------|---------------|-------------| | Current $I$ | Exponential decay | Maximum (on y-axis) | Zero | | Voltage $V$ | Exponential growth | Zero | Maximum (supply voltage) | | Charge $Q$ | Exponential growth | Zero | Maximum | **Key features of charging graphs:** - The p.d. and charge graphs have **identical shapes** - The current graph is an **exponential decay** curve - Current starts at maximum and decreases exponentially - P.d. and charge start at zero and increase to maximum values The graph below shows typical charging behavior (values normalized to percentages of maximum):

## Discharging a Capacitor - Capacitors are discharged through a **resistor** with **no power supply** present - Electrons flow back from the **negative plate** to the **positive plate** - Flow continues until there are equal numbers of electrons on each plate - No potential difference remains between the plates

Charging and discharging is commonly achieved using a switch that connects the capacitor between a power supply and a resistor

### Discharging Current, Voltage and Charge - At the start of discharge, the current is **large** (but in the **opposite direction** to when charging) - Current gradually falls to zero - As a capacitor discharges, **current, p.d. and charge all decrease exponentially** - The rate of decrease is proportional to the amount remaining - Like a phone battery draining - it loses 50% of its remaining charge in equal time intervals ### Discharging Graphs | Quantity | Graph Shape | Initial Value | Final Value | |----------|-------------|---------------|-------------| | Current $I$ | Exponential decay | Maximum (on y-axis) | Zero | | Voltage $V$ | Exponential decay | Maximum (initial voltage) | Zero | | Charge $Q$ | Exponential decay | Maximum (initial charge) | Zero | **Key features of discharge graphs:** - The shapes of **current, p.d. and charge** graphs are **identical** - Each graph shows **exponential decay** curves with **decreasing gradient** - Initial values (typically called $I_0$, $V_0$ and $Q_0$ respectively) start on the y-axis and decrease exponentially

Exponential decay means the rate of decrease is proportional to the quantity remaining

The graph below shows typical discharging behavior (values normalized to percentages of initial maximum):

## Effect of Resistance on Discharge Rate - The rate at which a capacitor discharges depends on the **resistance** of the circuit | Resistance | Current | Discharge Rate | Time to Discharge | |------------|---------|----------------|-------------------| | **High** | Decreases slowly | Charge flows slowly from plates | Takes **longer** | | **Low** | Decreases quickly | Charge flows quickly from plates | Discharges **faster** | - If resistance is **high**: - Current will be smaller - Charge flows from the capacitor plates more slowly - Capacitor takes **longer to discharge** - If resistance is **low**: - Current will be larger - Charge flows from the capacitor plates quickly - Capacitor will discharge **faster** The graph below compares discharge with different resistances (both starting from 10 V):

## Investigating the Charge & Discharge of a Capacitor ### Aim - Calculate the capacitance of a capacitor by investigating its discharge characteristics ### Variables - **Independent variable:** time, $t$ - **Dependent variable:** potential difference, $V$ - **Control variables:** - Resistance of the resistor - Current in the circuit ### Equipment | Apparatus | Purpose | |-----------|---------| | Switch | To switch between the charging and discharging circuit | | Capacitor | To measure the capacitance | | 10 kΩ Resistor | To discharge the capacitor | | Battery pack (power supply) | To provide the potential difference across the capacitor | | Voltmeter | To measure the potential difference across the capacitor | | Stopwatch | To measure the time taken for the capacitor to discharge | **Resolution of measuring equipment:** - Voltmeter = 0.1 V - Stopwatch = 0.01 s ### Method 1. Set up the circuit with the switch not connected to X or Y (no current flowing) 2. Set the battery pack to a potential difference of 10 V and use a 10 kΩ resistor - **Note:** The capacitor should initially be fully discharged 3. Charge the capacitor fully by placing the switch at point **X** - The voltmeter reading should equal the battery voltage (10 V) 4. Move the switch to point **Y** to begin discharge 5. Record the voltage reading every 10 s down to a value of 0 V - A total of 8-10 readings should be taken ### Analysing the Results - The potential difference across the capacitor during discharge follows: $$V = V_0 e^{-\frac{t}{RC}}$$ Where: - $V$ = p.d. across the capacitor (V) - $V_0$ = initial p.d. across the capacitor (V) - $t$ = time (s) - $e$ = exponential function - $R$ = resistance of the resistor (Ω) - $C$ = capacitance of the capacitor (F) **To find capacitance:** 1. Take the natural logarithm (ln) of both sides: $$\ln\left(\frac{V}{V_0}\right) = -\frac{t}{RC}$$ $$\ln(V) = -\frac{t}{RC} + \ln(V_0)$$ 2. Compare to straight line equation $y = mx + c$: - $y = \ln(V)$ - $x = t$ - gradient = $-\frac{1}{RC}$ - $c = \ln(V_0)$ 3. Plot a graph of $\ln(V)$ against $t$ and draw a line of best fit 4. Calculate the gradient (this should be negative) 5. The capacitance of the capacitor is: $$C = -\frac{1}{R \times \text{gradient}}$$ ### Evaluating the Experiment **Systematic Errors:** - If using a **digital voltmeter**, wait until the reading settles if it's switching between two values - If using an **analogue voltmeter**, reduce parallax error by reading at eye level to the meter - Make sure the voltmeter starts at zero to avoid a zero error **Random Errors:** - Use a resistor with **large resistance** so the capacitor discharges slowly enough for time to be taken accurately at p.d. intervals - Using a **datalogger** will provide more accurate results for the p.d. at a certain time - Reduces error in reaction time needed to stop the stopwatch at a certain p.d. - The experiment could be repeated by measuring the time for the capacitor to **charge** instead

Capacitors can retain charge after power is removed, which could cause an electric shock - fully discharge and leave for a few minutes

### Safety Considerations - Keep **water or any fluids** away from electrical equipment - Make sure no **wires or connections are damaged** - Ensure use of **appropriate fuses** to avoid a short circuit or a fire - Using a resistor with **too low resistance**: - Means the capacitor discharges too quickly for accurate measurements - Causes wires to become very hot due to high current - Creates a safety hazard