Download RC Circuits – Determining the Time Constant

RC Circuits – Determining the Time Constant
Purpose: To determine the time constant for an RC circuit.
Background: So far we have only discussed circuits where the voltages, resistances and
currents are constant. Now we will investigate RC circuits, where the voltage and
currents vary with time.
When a circuit with a resistor (R) and a capacitor (C) in series is closed, the capacitor is
initially uncharged and so the voltage across it is equal to 0. The voltage across the
resistor will be equal to the voltage of the emf. As the capacitor charges, there is an
increasing electric field between the capacitor plates and therefore an increasing voltage
across the capacitor. This means that the current in the circuit will decrease since the
voltage across the capacitor opposes the voltage across the batteries or power supply.
(You can think about this like two batteries that are put together with positive to positive
– the net result will be zero volts.) Eventually, the capacitor will be very nearly fully
charged and the current will effectively go to zero.
Mathematically, we can describe the voltage across the capacitor by the equation
VC (t) = Vo (1 – e-t/τ)
Where VC is the voltage across the capacitor (in volts)
Vo is the voltage across the batteries or power supply (in volts)
t is the time elapsed (in seconds)
τ is the time constant, RC (in seconds)
Procedure: Make an RC circuit such that the batteries or power supply, the resistor
and the capacitor are in series with each other but do not close the circuit until you are
ready to collect data. Be ready to measure the voltage across the capacitor. Close the
circuit so that the current flows and the capacitor will begin to charge. At time t = 0, the
voltage across the capacitor will be zero because the capacitor has not charged up yet.
Record the voltage every 30 seconds for about 10 - 15 minutes (more or less – just make
sure that the voltage across the capacitor is close to the voltage across the batteries or
power supply when you stop taking data).
In the event that you need to start over, you can simply discharge the capacitor by
running a wire from one end of the capacitor to the other and the capacitor will discharge
quickly (just a second). Set up the circuit again and start over.
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Data: Using the data table below or using Excel directly, make a data table that shows
the following data:
Voltage on power supply:
Time (s)
Resistor value:
Voltage (resistor) (V)
0
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Voltage (capacitor) (V)
Graph:
Make a graph of the voltage across the capacitor versus time.
Follow-Up Questions:
1.
What is the value of the horizontal asymptote on your graph? What does this
value signify?
2.
Using the equation above for the value of the voltage across the capacitor as a
function of time, find the voltage across the capacitor, VC, when t = τ. Your
answer will be as a function of the voltage across the batteries or power supply.
3.
What is the value of VC at t = τ for your experiment?
4.
At what time is this voltage reached (use your graph)?
5.
What are the units of RC?
6.
What is the numerical product of RC for the resistor and capacitor that you used?
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7.
How is the value of RC (question 6) related to your answer from question 4?
8.
If  = RC where  is the time constant, what does it mean to have a large or small
? How can you increase/decrease the time constant?
9.
Can you think of how a small (or large) time constant might be useful in
applications?
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