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The time constant and exponential decay questions
One number, R times C, sets the timescale for charging and discharging alike. After one time constant 37% is left, half is gone every 0.69 of one, and a log-linear plot flattens the whole exponential into a straight line you can measure.
18 original questions · 49 marks · the the time constant and exponential decay notes · Capacitance
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State the equation for the time constant of a capacitor-resistor circuit and explain what the time constant represents for a discharging capacitor.
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τ = RC (1); the time for the charge (or pd, or current) to fall to 1/e (about 37%) of its initial value (1).A 100 μF capacitor discharges through a 10 kΩ resistor. Calculate the time constant of the circuit.
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τ = RC = 10000 × 100 × 10−6 (1)
τ = 1.0 s (1)Write down the equation for the charge remaining on a capacitor at time t as it discharges through a resistor.
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Q = Q0e−t/RC, where Q0 is the initial charge (1).A capacitor charges through a resistor. State the percentage of its final charge that it has gained one time constant after charging begins.
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63% (the fraction 1 − 1/e) (1).A 47 μF capacitor discharges through a 2.2 MΩ resistor. Calculate the time constant of the circuit.
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τ = RC = 2.2 × 106 × 47 × 10−6 (1)
τ = 103 s (1)A 470 μF capacitor discharges through a 2.2 kΩ resistor. Calculate the time constant and state the fraction of the initial charge that remains after one time constant.
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τ = RC = 2200 × 470 × 10−6 (1)
τ = 1.03 s (1)
After one time constant, Q/Q0 = 1/e ≈ 0.37 (37%) (1).A 100 μF capacitor is charged to 12 V and then discharged through a 47 kΩ resistor. Calculate the potential difference across the capacitor 5.0 s after the discharge begins.
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τ = RC = 47000 × 100 × 10−6 = 4.7 s (1)
V = V0e−t/RC = 12 × e−5.0/4.7 (1)
V = 4.14 V (1)For the circuit in the previous question, calculate the time taken for the potential difference to halve.
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T½ = 0.69RC = 0.69 × 4.7 (1)
T½ = 3.24 s (1)A timing circuit requires a time constant of 2.0 s using a 220 μF capacitor. Calculate the resistance required.
In required practical 9 a student discharges a capacitor through a resistor while logging the pd across it. The pd falls from 6.0 V to 2.2 V in 15 s. Show that the time constant of the circuit is about 15 s.
A capacitor discharges through a 33 kΩ resistor. The pd across it halves every 4.6 s. Determine the capacitance of the capacitor.
A capacitor discharges through a resistor with a time constant of 3.0 s. Determine the time taken for the potential difference across it to fall from 20 V to 5.0 V.
A 470 μF capacitor is charged from a 9.0 V supply through a 10 kΩ resistor. Calculate the potential difference across the capacitor 10 s after charging begins.
A student plots ln(V) against time for a discharging capacitor. Explain how the time constant can be determined from this graph.
A student uses required practical 9 to test a capacitor labelled 470 μF with a tolerance of ±20%. The capacitor is discharged through a 10 kΩ resistor, and the graph of ln(V) against t is a straight line of gradient −0.185 s−1. Deduce whether the capacitor is within its stated tolerance.
A capacitor discharges through a resistor. Show that the energy stored falls to about 14% of its initial value after one time constant.
The manual for a high-voltage supply states that its reservoir capacitor must be left to discharge until less than 1.0% of the initial charge remains before the case is opened. The discharge time constant is 4.0 s. Determine the minimum waiting time.
Describe an experiment to determine the capacitance of an unmarked capacitor by discharging it through a resistor of known resistance. Include the measurements you would take and explain how you would analyse the data to obtain the capacitance.
The same practice on paper: the printable workbook for this topic, questions and a worked answer book.
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