AQA A Level Physics Capacitor Charge & Discharge PDF

Head to www.savemyexams.com for more awesome resources AQA A Level Physics Your notes 7.7 Capacitor Charge & Discharge Contents 7.7.1 Charge & Discharge Graphs 7.7.2 The Time Constant 7.7.3 Charge & Discharge...

Head to www.savemyexams.com for more awesome resources AQA A Level Physics Your notes 7.7 Capacitor Charge & Discharge Contents 7.7.1 Charge & Discharge Graphs 7.7.2 The Time Constant 7.7.3 Charge & Discharge Equations 7.7.4 Required Practical: Charging & Discharging Capacitors Page 1 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources 7.7.1 Charge & Discharge Graphs Your notes Capacitor Charge & Discharge Graphs Charging Capacitors are charged by a power supply (eg. a battery) When charging, the electrons are pulled from the plate connected to the positive terminal of the power supply Hence the plate nearest the positive terminal is positively charged They travel around the circuit and are pushed onto the plate connected to the negative terminal Hence the plate nearest the negative terminal is negatively charged As the negative charge builds up, fewer electrons are pushed onto the plate due to electrostatic repulsion from the electrons already on the plate When no more electrons can be pushed onto the negative plate, the charging stops A parallel plate capacitor is made up of two conductive plates with opposite charges building up on each plate At the start of charging, the current is large and gradually falls to zero as the electrons stop flowing through the circuit Page 2 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources The current decreases exponentially This means the rate at which the current decreases is proportional to the amount of current it has left Your notes Since an equal but opposite charge builds up on each plate, the potential difference between the plates slowly increases until it is the same as that of the power supply Similarly, the charge of the plates slowly increases until it is at its maximum charge defined by the capacitance of the capacitor Graphs of variation of current, p.d and charge with time for a capacitor charging through a battery The key features of the charging graphs are: The shapes of the p.d. and charge against time graphs are identical The current against time graph is an exponential decay curve The initial value of the current starts on the y axis and decreases exponentially The initial value of the p.d and charge starts at 0 up to a maximum value Discharging Capacitors are discharged through a resistor with no power supply present The electrons now flow back from the negative plate to the positive plate until there are equal numbers on each plate and no potential difference between them Charging and discharging is commonly achieved by moving a switch that connects the capacitor between a power supply and a resistor Page 3 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Your notes The capacitor charges when connected to terminal P and discharges when connected to terminal Q At the start of discharge, the current is large (but in the opposite direction to when it was charging) and gradually falls to zero As a capacitor discharges, the current, p.d and charge all decrease exponentially This means the rate at which the current, p.d or charge decreases is proportional to the amount of current, p.d or charge it has left The graphs of the variation with time of current, p.d and charge are all identical and follow a pattern of exponential decay Page 4 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Graphs of variation of current, p.d and charge with time for a capacitor discharging through a resistor The key features of the discharge graphs are: Your notes The shape of the current, p.d. and charge against time graphs are identical Each graph shows exponential decay curves with decreasing gradient The initial values (typically called I0, V0 and Q0 respectively) start on the y axis and decrease exponentially The rate at which a capacitor discharges depends on the resistance of the circuit If the resistance is high, the current will decrease and charge will flow from the capacitor plates more slowly, meaning the capacitor will take longer to discharge If the resistance is low, the current will increase and charge will flow from the capacitor plates quickly, meaning the capacitor will discharge faster Exam Tip Make sure you're comfortable with sketching and interpreting charging and discharging graphs, as these are common exam questionsRemember that conventional current flow is in the opposite direction to the electron flow Page 5 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Properties of Capacitor Discharge Graphs From electricity, the charge is defined: Your notes ΔQ = I Δt Where: I = current (A) ΔQ = change in charge (C) Δt = change in time (s) This means that the area under a current-time graph for a charging (or discharging) capacitor is the charge stored for a certain time interval The area under the I-t graph is the total charge stored in the capacitor in the time interval Δt Rearranging for the current: This means that the gradient of the charge-time graph is the current at that time Page 6 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Your notes The gradient of a discharging and charging Q-t graph is the current In the discharging graph, this is the discharging current at that time In the charging graph, this is the charging current at that time To calculate the gradient of a curve, draw a tangent at that point and calculate the gradient of that tangent Page 7 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Worked example Your notes The graph below shows how the charge stored on a capacitor with capacitance C varies with time as it discharges through a resistor. Calculate the current through the circuit after 4 s. Step 1: Draw a tangent at t = 4 Page 8 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Your notes Step 2: Calculate the gradient of the tangent to find the current I Page 9 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources 7.7.2 The Time Constant Your notes The Time Constant The time constant of a capacitor discharging through a resistor is a measure of how long it takes for the capacitor to discharge The definition of the time constant is: The time taken for the charge, current or voltage of a discharging capacitor to decrease to 37% of its original value Alternatively, for a charging capacitor: The time taken for the charge or voltage of a charging capacitor to rise to 63% of its maximum value 37% is 0.37 or 1 / e (where e is the exponential function) multiplied by the original value (I0, Q0 or V0) This is represented by the Greek letter tau, τ , and measured in units of seconds (s) The time constant provides an easy way to compare the rate of change of similar quantities eg. charge, current and p.d. It is defined by the equation: τ = RC Where: τ = time constant (s) R = resistance of the resistor (Ω) C = capacitance of the capacitor (F) For example, to find the time constant from a voltage-time graph, calculate 0.37V0 and determine the corresponding time for that value Page 10 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Your notes The time constant shown on a charging and discharging capacitor The time to half, t1/2 (half-life) for a discharging capacitor is: The time taken for the charge, current or voltage of a discharging capacitor to reach half of its initial value This can also be written in terms of the time constant: t1/2 = 0.69 τ = 0.69RC Worked example A capacitor of 7 nF is discharged through a resistor of resistance R. The time constant of the discharge is 5.6 × 10-3 s.Calculate the value of R. Page 11 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Your notes Exam Tip Remember to check the context of an exam question, i.e., whether the capacitor is charging or discharging. The definition of the time constant depends on it! For a charging capacitor, the time constant refers to the time taken to reach 63% of its maximum potential difference or charge stored For a discharging capacitor, the time constant refers to the time take to discharge to 37% of its initial potential difference or charge stored Page 12 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources 7.7.3 Charge & Discharge Equations Your notes Capacitor Discharge Equation The time constant is used in the exponential decay equations for the current, charge or potential difference (p.d) for a capacitor discharging through a resistor These can be used to determine the amount of current, charge or p.d left after a certain amount of time for a discharging capacitor This exponential decay means that no matter how much charge is initially on the plates, the amount of time it takes for that charge to halve is the same The exponential decay of current on a discharging capacitor is defined by the equation: Where: I = current (A) I0 = initial current before discharge (A) e = the exponential function t = time (s) RC = resistance (Ω) × capacitance (F) = the time constant τ (s) This equation shows that the smaller the time constant τ, the quicker the exponential decay of the current when discharging Also, how big the initial current is affects the rate of discharge If I0 is large, the capacitor will take longer to discharge Note: during capacitor discharge, I0 is always larger than I, as the current I will always be decreasing Page 13 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Your notes Values of the capacitor discharge equation on a graph and circuit The current at any time is directly proportional to the p.d across the capacitor and the charge across the parallel plates Therefore, this equation also describes the charge on the capacitor after a certain amount of time: Where: Q = charge on the capacitor plates (C) Q0 = initial charge on the capacitor plates (C) As well as the p.d after a certain amount of time: Where: V = p.d across the capacitor (C) V0 = initial p.d across the capacitor (C) The Exponential Function e The symbol e represents the exponential constant, a number which is approximately equal to e = 2.718... On a calculator, it is shown by the button ex The inverse function of ex is ln(y), known as the natural logarithmic function This is because, if ex = y, then x = ln (y) Page 14 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources The 0.37 in the definition of the time constant arises as a result of the exponential constant, the true definition is: Your notes Worked example The initial current through a circuit with a capacitor of 620 µF is 0.6 A.The capacitor is connected across the terminals of a 450 Ω resistor.Calculate the time taken for the current to fall to 0.4 A. Page 15 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Your notes Exam Tip The equation for Q will be given on the data sheet, however you will be expected to remember that it is similar for I and V. Page 16 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Capacitor Charge Equation When a capacitor is charging, the way the charge Q and potential difference V increases stills shows Your notes exponential decay Over time, they continue to increase but at a slower rate This means the equation for Q for a charging capacitor is: Where: Q = charge on the capacitor plates (C) Q0 = maximum charge stored on capacitor when fully charged (C) e = the exponential function t = time (s) RC = resistance (Ω) × capacitance (F) = the time constant τ (s) Similarly, for V: Where: V = p.d across the capacitor (V) V0 = maximum potential difference across the capacitor when fully charged (V) The charging equation for the current I is the same as its discharging equation since the current still decreases exponentially The key difference with the charging equations is that Q0 and V0 are now the final (or maximum) values of Q and V that will be on the plates, rather than the initial values Worked example A capacitor is to be charged to a maximum potential difference of 12 V between its plate. Calculate how long it takes to reach a potential difference 10 V given that it has a time constant of 0.5 s. Page 17 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Your notes Page 18 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Your notes Exam Tip Make sure you’re confident in rearranging equations with natural logs (ln) and the exponential function (e) for both charging and discharging equations. To refresh your knowledge of this, have a look at the AS Maths revision notes on Exponentials & Logarithms. Page 19 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources 7.7.4 Required Practical: Charging & Discharging Capacitors Your notes Required Practical: Charging & Discharging Capacitors Aim of the Experiment The overall aim of this experiment is to calculate the capacitance of a capacitor. This is just one example of how this required practical might be carried out Variables Independent variable = time, t Dependent variable= potential difference, V Control variables: Resistance of the resistor Current in the circuit Equipment List Resolution of measuring equipment: Voltmeter = 0.1 V Stopwatch = 0.01 s Method Page 20 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Your notes 1. Set up the apparatus like the circuit above, making sure the switch is not connected to X or Y (no current should be flowing through) 2. Set the battery pack to a potential difference of 10 V and use a 10 kΩ resistor. The capacitor should initially be fully discharged 3. Charge the capacitor fully by placing the switch at point X. The voltmeter reading should read the same voltage as the battery (10 V) 4. Move the switch to point Y 5. Record the voltage reading every 10 s down to a value of 0 V. A total of 8-10 readings should be taken An example table might look like this: Page 21 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Your notes Analysing the Results The potential difference (p.d) across the capacitance is defined by the equation: Where: V = p.d across the capacitor (V) V0 = 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) Rearranging this equation for ln(V) by taking the natural log (ln) of both sides: Page 22 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Comparing this to the equation of a straight line: y = mx + c y = ln(V) x=t Your notes gradient = -1/RC c = ln(V0) 1. Plot a graph of ln(V) against t and draw a line of best fit 2. Calculate the gradient (this should be negative) 3. The capacitance of the capacitor is equal to: Evaluating the Experiment Systematic Errors: If a digital voltmeter is used, wait until the reading is settled on a value if it is switching between two If an analogue voltmeter is used, reduce parallax error by reading the p.d at eye level to the meter Make sure the voltmeter starts at zero to avoid a zero error Random Errors: Use a resistor with a large resistance so the capacitor discharges slowly enough for the 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. This will reduce the error in the speed of the reflex 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 Safety Considerations Keep water or any fluids away from the electrical equipment Page 23 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Make sure no wires or connections are damaged and contain appropriate fuses to avoid a short circuit or a fire Using a resistor with too low a resistance will not only mean the capacitor discharges too quickly but Your notes also that the wires will become very hot due to the high current Capacitors can still retain charge after power is removed which could cause an electric shock. These should be fully discharged and removed after a few minutes Worked example A student investigates the relationship between the potential difference and the time it takes to discharge a capacitor. They obtain the following results: The capacitor is labelled with a capacitance of 4200 µF. Calculate: (i) The value of the capacitance of the capacitor discharged. (ii) The relative percentage error of the value obtained from the graph and this true value of the capacitance. Step 1: Complete the table Add an extra column ln(V) and calculate this for each p.d Page 24 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Your notes Step 2: Plot the graph of ln(V) against average time t Make sure the axes are properly labelled and the line of best fit is drawn with a ruler Step 3: Calculate the gradient of the graph Page 25 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers  Head to www.savemyexams.com for more awesome resources Your notes The gradient is calculated by: Step 4: Calculate the capacitance, C Step 5: Calculate the relative percentage error of the value obtained Page 26 of 26 © 2015-2024 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers