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Transcript
23 Capacitors Support AQA Physics Capacitance graphs Specification reference 3.7.4 MS 0.1, 0.3, 0.5, 2.2, 2.3, 2.4, 3.1, 3.8, 3.10, 3.11, 3.12 Introduction Capacitors are electronic components that store charge and so can be used in time delay circuits. For example, as you open your front door there is a delay (giving you a chance to enter the correct code) before the burglar alarm goes off. Time delay circuits work with a combination of a capacitor and a resistor. The larger the values of resistance and capacitance in the circuit, the longer the time delay will be. Figure 1 Capacitors come in different shapes and sizes. They can be two parallel plates with air or another insulating material called a dielectric between them, or cylindrical where the two ‘plates’ and the dielectric are effectively rolled up rather like a ‘swiss roll’ − visualise this as the ‘cake’ being the plates and the ‘filling’ the dielectric and charges sandwiched between. Capacitors can be charged and discharged and the graphs produced are exponential growth or decay. © Oxford University Press 2016 http://www.oxfordsecondary.co.uk/acknowledgements This resource sheet may have been changed from the original 1 23 Capacitors Support AQA Physics Learning outcomes After completing this worksheet you should be able to: explain the meaning of capacitance use the capacitance equation explain the similarities and differences between graphs showing charging and discharging of a capacitor and be able to sketch them all explain the meaning of the time constant and be able to find it from the equations and from a graph find the energy stored in a capacitor from the equations and also from a graph. Background Capacitance Units: charge (coulombs, C), capacitance (farads, F) potential difference (volts, V) Capacitance is often quoted in microfarads (μF) 10–6 F Figure 2 Energy stored in capacitor = 1 1 1 Q2 QV = CV 2 = 2 2 2 C Capacitor discharge Q Q0 e t – RC Figure 3 © Oxford University Press 2016 V V0 e Figure 4 t – RC I I0 e t – RC Figure 5 http://www.oxfordsecondary.co.uk/acknowledgements This resource sheet may have been changed from the original 2 23 Capacitors Support AQA Physics All curves in Figures 1 to 3 have the same shape. RC is known as the time constant time taken for charge to drop to e−1 0.37 of initial value After a time RC the voltage, charge, and current will all have been reduced to 0.37 of their original value. As the discharge of a capacitor is exponential you could be asked to plot a ln graph. – t V V0 e RC Taking logs of both sides ln V0 t RC Comparing with y mx c Intercept on ln V axis ln V0 ln V ln V 0 - Figure 6 Gradient − 1 RC Charging a capacitor V V0 (1– e t – RC Q Q0 (1– e ) Figure 7 Figure 8 t – RC I I0 e ) Figure 9 The Q and V curves in figures 7 and 8 have same shape but I in Figure 9 is opposite. As a capacitor is charged, charge is flowing so Q increases but due to electrostatic repulsion, current I ( charge flowing per second) decreases. Current 0 when capacitor is fully charged. I = t – RC Q = gradient of Q curve. t © Oxford University Press 2016 http://www.oxfordsecondary.co.uk/acknowledgements This resource sheet may have been changed from the original 3 23 Capacitors Support AQA Physics Worked examples Example 1: Energy stored area under V–Q graph When fully charged by a 20 V DC supply a capacitor carries a charge of 5.0 μC. Calculate: a the capacitance b the energy stored in the capacitor. Q 5.0 μC Step 1 Use C = V 20 V C? Energy ? Q and substitute in values. V 5.0 ´10-6 20 0.25 × 10−6 F or 0.25 μF Step 2 Use Energy stored QV (or you could use CV 2 if you wish). 1 × 5.0 × 10−6 × 20 2 5.0 × 10−5 J OR The information from the same question could have been represented graphically as shown below. Figure 10 In this case, a Capacitance 1 gradient 5.0 ´10-6 20 0.25 × 10−6 F or 0.25 μF = b Energy stored area under graph 1 × 20 × 5.0 × 10−6 2 5.0 × 10−5 J © Oxford University Press 2016 http://www.oxfordsecondary.co.uk/acknowledgements This resource sheet may have been changed from the original 4 23 Capacitors Support AQA Physics Example 2: Discharging Q and V graphs In the circuit in Figure 11 a fully charged capacitor has a potential difference of 6.0 V across it. The switch is then closed. Calculate: a b c d the time constant the pd across the capacitor after 20 s the charge on the capacitor after 20 s. Show all of this information graphically. (You will need to find initial charge Q0 and calculate the values of V and Q when t time constant.) Figure 11 a R 100 kΩ C 500 μF Step 1 Time constant R × C 100 × 103 × 500 × 10−6 50 s b V0 6.0 V t 20 s RC 50 s V decreases exponentially Step 2 Use V V0 e t – RC 6.0 e . 20 – 50 6.0 e −0.4 4.0 V c At t 20 s V 4.0 V C 500 × 10−6 F Step 3 Use Q CV. 500 × 10−6 × 4.0 2.0 × 10−3 C d Graphically Initially charge Q0 CV 500 × 10−6 × 6.0 3.0 C When t RC 50 s, V 0.37 V0, and Q 0.37 Q0 Figure 12 © Oxford University Press 2016 Figure 13 http://www.oxfordsecondary.co.uk/acknowledgements This resource sheet may have been changed from the original 5 23 Capacitors Support AQA Physics Example 3: ln graph A 10 μF capacitor is charged to a potential difference of 12 V and then discharged through a 100 kΩ resistor. a Calculate the time taken before the potential difference across the capacitor drops to 1 V. b Show this information graphically. V0 12 V V 1 V R 100 kΩ C 10 μF t ? Step 1 Calculate RC 100 × 103 × 10 × 10−6 1 s Step 2 Use V Vo e t – RC –t 1 12 e 1 Step 3 Take ln of both sides ln 1 ln 12 − t Step 4 ln 1 0 t ln 12 t 2.5 s (to two significant figures) Figure 14 Questions 1 a Calculate the charge stored on the plates of a capacitor of capacitance 8 μF when it is connected to a 2.0 V battery. (2 marks) b Calculate the energy stored in the capacitor. (2 marks) c Show this information graphically. (2 marks) © Oxford University Press 2016 http://www.oxfordsecondary.co.uk/acknowledgements This resource sheet may have been changed from the original 6 23 Capacitors Support AQA Physics 2 A 1000 μF capacitor is charged to 5000 V, disconnected from the power supply and then allowed to discharge through a 100 kΩ resistor. Calculate the time taken for the voltage across the capacitor to fall to: a 200 V (3 marks) b 100 V (2 marks) c 1V (2 marks) 3 A 2500 μF capacitor is charged through a 1 kΩ resistor by a 12 V power supply. Calculate the voltage across the capacitor after 5 s. (2 marks) Graphical tasks A 200 mF capacitor is charged to 10 V and then discharged through a 250 kΩ resistor. a Calculate the time constant RC. b Calculate the pd across the capacitor at intervals of 10 s and complete the second row of Table 1 using V V0 e t – RC t/s 0 10 20 V/V 10 8.2 6.7 ln (V / V) 2.303 I /μA 40 Q /mC 2.0 . 30 40 50 60 70 Table 1 © Oxford University Press 2016 http://www.oxfordsecondary.co.uk/acknowledgements This resource sheet may have been changed from the original 7 23 Capacitors Support AQA Physics c d e f g Draw a V against t graph or produce one using a spreadsheet. Mark on the graph the value of V when t RC. Complete the third row of the table. Draw a ln V against t graph or produce one using Excel. Determine the gradient. What does this represent? V0 and complete the fourth row of the table. R i Calculate Q (remember Q0 CV0). j Draw a Q against t graph or produce one using Excel. k Determine the gradient at t 10 s. What does this represent? l Draw a I against t graph or produce one using Excel. h Calculate I (remember I0 © Oxford University Press 2016 http://www.oxfordsecondary.co.uk/acknowledgements This resource sheet may have been changed from the original 8