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Loughborough College – International Foundation Studies. Physics Adapted from Foundation Studies, Department of Physics, Loughborough University Semester 1 Experiment 2: Capacitor Discharge Aims Experimental: Investigate the discharge of a capacitor with time. Practical: Use of an oscilloscope to make repeated voltage readings. Theoretical: To understand and calculate exponential decay and dc circuits. Safety Normal laboratory safety procedures apply. There are no special precautions needed for this experiment. Background and purpose When a charged capacitor is discharged through a fixed resistor, charge flows through the resistor from one plate of the capacitor to the other. The p.d. across the capacitor decreases as it discharges. This experiment is intended to investigate how the p.d. varies with time, and to give experience in working with logarithmic formulae to calculate the time constant for a capacitor / resistor circuit. The oscilloscope is being used here simply as a DC Voltmeter, not to get a discharge curve directly onto the screen. Procedure R1 R2 Connect to Oscilloscope Y input 1) Connect circuit as shown above to oscilloscope CH1. Note that R1 and R2 have identical values 2) Set the channel on the oscilloscope to ground and adjust the trace until it coincides with the lowest graticule line. Switch the channel to DC and adjust the scale so that there is a maximum displacement on the screen (checking that the ground position has not moved). When adjusting the X-gain you must ensure that the variable gain is set to “calib” 3) Set R2 switch to “open” so R2 is not in the circuit. Capacitor Discharge Semester 1 Page 1 of 3 Loughborough College – International Foundation Studies. Physics Adapted from Foundation Studies, Department of Physics, Loughborough University 4) Next set switch to Position A to charge the capacitor. 5) Reset the switch to Position B to discharge the capacitor through the resistor R1 6) Measure the capacitor p.d. using the oscilloscope at regular intervals. NOTE: You should undertake a preliminary run through the experiment up to the end of point 6 to determine the appropriate oscilloscope settings and also a suitable time interval for the measurements. 7) Recharge the capacitor once you have set the oscilloscope gain to a suitable value. 8) Take a set of readings of the p.d. during discharge, as described in 6 above. 9) Next set switch to “closed” to bring R2 into the circuit, in parallel with resistor R1. What does this do to the overall resistance in the circuit? 10) Repeat the procedure with this new resistor arrangement. 11) For each of the two sets of measurements, plot a graph of capacitor p.d. against time. Data Analysis The discharge of a capacitor is an exponential decay, from the starting p.d., V0, to a generalised p.d., V, at time, t. The mathematical expression for this is: 𝑉 = 𝑉0 𝑒 −𝑡 / 𝜏 (1) As the exponential function can operate only on a pure number, the parameter τ must have the dimensions of time; it is known as the time constant. (The “τ” symbol is the Greek letter “tau”) The time constant is the characteristic time over which an exponential function changes. For a circuit containing a resistor and a capacitor such as the one here, the time constant is given by: τ = RC (2) Note that when t= τ, the p.d. will be V0/e and as e is 2.718281828, the time constant is approximately the time taken for the voltage to fall to one third. (Check this for yourself) Another common measure of exponential decay is the half-life, T½. This is the time taken for the quantity to fall to one half of its starting value. The value will halve again to one quarter after another half-life and so on. Simple mathematics will readily show the relationship between the half-life and the time constant to be: Capacitor Discharge Semester 1 Page 2 of 3 Loughborough College – International Foundation Studies. Physics Adapted from Foundation Studies, Department of Physics, Loughborough University 𝜏= 𝑇1/2 ln 2 (3) For each graph do the following: 12) Determine how long the capacitor took to discharge to a) 0.5 and b) 0.25 of the initial p.d. in each case and estimate the half-life. 13) Calculate the starting voltage divided by e and from the graph, make a rough estimate of τ. 14) Obtain a more accurate estimate of τ by doing the following: Rearranging the expression for p.d. we get: 𝑉 𝑉0 = 𝑒 −𝑡 / 𝜏 Taking natural logarithms gives: ln( 𝑉 ) 𝑉0 = −𝑡 𝜏 and find the time constant. 15) Using your values for τ and T½, verify equations (2) and (3) within experimental error. Capacitor Discharge Semester 1 Page 3 of 3