Download Semester 1 Experiment 2 Capacitors Locoll version

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
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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
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