1. Read the accompanying text and
become familiar with the operations of the applet. Flip the switch so that the EMF
(battery) is connected to the circuit. In this situation, we have a simple RC series
circuit. Press the "Reset" button so that the capacitor is initially
uncharged.
(a) Starting at the positive terminal of the EMF source, sum the voltage drops around
the circuit. Denote the EMF with the variable V. Do not use numerical
values for any physical properties at this point.
2. For those having an understanding of basic differential equations,
answer the following. (For others, skip to Question 3).
(a) From your answer to the previous question, and the fact that the capacitor is
initially uncharged, show that the charge Q on the capacitor evolves with time according
to the relationship:
Q(t) = CV- D Exp[-t/(RC)],
where D is a constant to be determined and Exp[-t/(RC)] means "the exponential of
-t/RC."
(b) Using the fact that the capacitor is uncharged, show that this equation reduces to
Q(t) = CV{1 - Exp[-t/(RC)]}.
3. The left-hand side of the above equation is the charge
separation of the capacitor, which can also be written as Q(t) = CV(t), where V(t) is the
voltage drop across the capacitor as a function of time. In other words
CV(t) = CV{1 - Exp[-t/(RC)]}.
(Yes, we could cancel the capacitance values on both sides, but we want to leave the C
on both sides of the equation since the applet plots the time evolution of the product CV,
and not just the voltage V.) We will now test this relationship and see what it has to say
theoretically about the time evolution of the voltage drop across the capacitor
plates.
(a) From this relationship, verify that at t = 0 the voltage drop across
the plates of the capacitor will be 0.
(b) What does this fact say about the charge separation across the plates at t
= 0?
(c) What is the maximum value the product CV(t) will achieve, no matter how
long the circuit is connected? (Hint: Let t become infinite.)
(d) How long will it take the charge to build to half the maximum value obtained in
previous question [Question 3(c)]? This time is called the time constant
of the circuit.
4. Now let's verify the charge-time evolution relationship numerically.
Notice that in the upper half of the applet is a plot of CV versus t.
(a) Press the "Start" button and describe the behavior of
the plot. On a sheet of graph paper, reconstruct the plot, complete with labeled
axes and numbered tic marks. (Note: The plot line needs
to start at 0 on the vertical CV axis since the capacitor is initially uncharged. If
it does not, press "Reset" and start over. )
(b) Notice that CV tend towards a maximum value. What is this
value?
(c) Is this maximum value of CV what you expected theoretically?
(d) On your plot, indicate the time at which CV reaches half its maximum
value. Write down on the plot this value of time. (Remember, we call this
time the time constant).
(e) Is this value close to what you expected theoretically?
How far off is it? (Use a percent error or percent difference.)
5. Now I want you to answer some conceptual questions.
(a) When the applet is running, why does the current drop in the circuit? (In
other words, why does the charge slow down?) Answer in terms of the charge polarity
on the plates of the capacitor.
(b) The height of the three light-blue vertical bars represent the voltage drop values
across each component of the circuit. Furthermore, these values are also shown
numerically. Explain how these three values are related. As the applet
runs, explain what happens to these values and why.
(c) Notice that charge flows through the circuit, even though the capacitor is, in
essence, a break in the circuit. Explain in terms of the flow of electrons and the
charge separation on the capacitor plates.
(d) When the switch is flipped back such that the battery is taken out of the circuit,
why does the capacitor discharge?
6. Now it is time to discharge the circuit.
(a) Once the current slows to a creep, use the mouse to flip the switch such that
the battery is shorted. (Note: Do not press "Stop"
during this process. ) What happens? Explain why in terms of the
charge separation on the capacitor and the flow of charge in the circuit.
(b) Ultimately, what will happen to the charge separation on the capacitor?
(c) Draw the resulting plot on your graph paper. Compare the two lines
(charging and discharging).
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