Shark Fins & Differential Equations

Physics 241 Lab – Matt Leone
Week 7: Shark Fins & Differential Equations
Leone@physics.arizona.edu (email preferred), PAS 376, o. 520-621-6819
Office Hours: M & W 11:00-11:50, or by appointment. Consultation Room (PAS 372): F 12:00-12:50
http://bohr.physics.arizona.edu/~leone/phy241/phys241lab.html
General Comments:
• Today we study DIRECT CURRENT charging and discharging of capacitors.
• You will be gently introduced to the mathematics of differential equations.
Lab 7 – Summary:
The physics concepts of this lab are very simple, and demonstrating them experimentally
straightforward. We could get done in less than an hour if we stuck to the experiment. Therefore, it is
the perfect time to explore some advanced math: differential equations.
Lab 7 – Matt’s discussion on board: now on the paper!
We will study the very simple circuit shown below. It is called the RC circuit:
In this circuit, Q(0) is the initial amount of charge on each plate of the capacitor (+ or –). R is the
resistance of the resistor in the circuit. C is the capacitance of the capacitor. In SI units, this is
typically very small, i.e. 1x10 -6 Farads is quite large for normal circuitry. When the switch is thrown,
the charge flows from one capacitor plate through the resistor to the other capacitor plate until the
charges are completely neutralized.
When you discharge a capacitor through the above circuit, the charge on each capacitor plate
decays exponentially in time,
Q(t) = Q(0)e " t
RC
.
Since you know that "V = Q , you can rewrite this equation by dividing both sides by C:
C
! V(t) = V(0)e " t
RC
.
Incidentally, when you charge a capacitor, the charge accumulates on the plates with
!
#
Q(t) = Q MAX
1" e " t &
#
RC
% ( or alternativelyV(t) = V APPLIED
1" e " t &
RC
% (.
$ ! '
$ '
In other words, you must apply the voltage long enough to charge Q MAX on the capacitor plate.
!
!
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Write your answers directly onto these pages and be sure to hand them in for credit with your
report!
Lab 7 – Theory Worksheet – Differential Equations – Together
1. Can you use a math model? Suppose I told you that the equation describing the time
dependence of the electric potential difference across a capacitor in a circuit was
"V (t) = "V (0)e # t $
.
Now imagine that you measured "V (0) = 3 volts and "V (10 seconds) =1 volt . Calculate the
time constant τ. You will see later in this activity that " = RC, and since you can measure R and
calculate τ, you can find that capacitance of the capacitor, C.
!
[Your work]:
!
!
2. In what follows, we will derive a differential equation to model an RC circuit. Then we will
solve it. I cannot stress the importance of these two steps. Most physicists make their living by
• Creating a differential equation to model reality, and
• Solving the differential equation to make predictions about reality.
Really, that is pretty much what we do!
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Construction of the differential equation (the physics). When you go hiking on a mountain and
return to your starting location, you will have a net change in height of zero, "( height) = 0. By
analogy, when you add up the change in electric potential around an RC circuit and end where
you began, you should find that "V = 0.
0 = "V resistor
+"V capacitor
.
!
You can then use Ohm’s law for the resistor and the definition of capacitance to get:
! 0 = RI + Q(t) capacitor
.
C
!
Now, what is the current through the resistor? It is the charge per time passing through
the resistor. So we can write
! 0 = R dQ(t) Resistor
+ Q(t) capacitor
.
dt C
Since the charges that flow onto or off of the capacitor plates must do so at the same
rate as through the resistor (or they will bunch up inside the resistor - nonsensical) we can
assume that
dQ(t) Resistor
dt
! = dQ(t) Capacitor
dt
so that we have
0 = R dQ(t) capacitor
+ Q(t) capacitor
.
dt C
!
This is a differential equation, an equation that contains derivatives!
a. What is the independent variable in this equation?
!
b. What is the dependent variable?
c. If the solution to an algebraic equation is one or more points, what is the solution to a
differential equation supposed to be?
d. What is the highest order time derivative in this equation? So how many functionsolutions
will we have? (Compare to linear and quadratic algebraic equations.)
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Restating the differential equation:
0 = R dQ(t)
dt
+ Q(t)
C
3. Solving the differential equation to find Q(t) (the math).
We have two physical situations in our lab experiments today, charging and discharging
the capacitor. We will only examine discharging
!
the capacitor because the math is pretty much
the same.
Discharging the capacitor occurs when you close the switch and let charges flow.
Rearranging the differential equation a little bit gives:
0 = dQ(t) + Q(t) dQ(t)
or rather
= - Q(t)
RC
dt RC dt
The best method for solving differential equations is guessing. We want a function Q(t)
such that when you take its time derivative, you get the same function back with a negative
1/RC. A function that has this property is an exponential function.
!
i. Allow me to guess the function-solution for you: Q(t) = e " t
RC
. Plug this guess solution into
the differential equation and show that it is in fact a solution.
[your work]
!
ii. What value does your equation give for Q(0) at the beginning of the discharge?
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Restating the discharging differential equation:
dQ(t)
dt
= - Q(t)
RC
iii. What if your capacitor had an initial charge, Q(0), that did not equal 1? Try the solution
Q(t) = Q(0)e " t
RC
and check that
!
1. it gives the correct initial value, and
!
2. it still satisfies the differential equation.
This step of solving differential equations is called taking account of the boundary
conditions.
iv. Make a quick and tiny sketch of this solution, Q(t) vs. t.
v. What do you get for
Lim
t "#
Q(t) and does this make sense?
!
vi. Does discharging the capacitor occur faster or slower if you increase the capacitance of the
capacitor. Use your graphing calculator to find out!
vii. Does discharging the capacitor occur faster or slower if you increase the resistance of the
resistor. Use your graphing calculator to find out!
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Restating the solution in terms of voltage: V(t) = V(0)e " t
RC
.
THIS IS THE EQUATION YOU WILL BE TESTING TODAY!
Lab 7 – Together Section of Lab – reminder: your solutions on these pages are worth serious
points today!
1. Experiment 1: Measuring the RC circuit, V vs. t.
a. Set your function generator to create a square wave with a voltage alternating between
V MIN = 0 Volts and V MAX = 3 Volts with a frequency of 1,000 Hz. Do this by 1 st setting the
frequency, then setting the wave to oscillate between +1.5 V and -1.5V, then use the DC
offset to shift your signal to have V MIN = 0 Volts. What do you see on the oscilloscope?
(YOU NEED TO USE LABELED AXIIS WITH LABELED HASH MARKS!!!)
!
b. Now set up an RC circuit with R=1 kΩ and C=0.1 µF in series with the same square wave
from the previous question. Then answer the following questions.
i. During the time interval that the square wave is at +3 Volts, is
the capacitor being charged or discharged?
ii. During the time interval that the square wave is at +0 Volts, is
the capacitor being charged or discharged?
iii. Which component must you measure to determine the total
current of the circuit?
iv. I want you to measure the voltage across the capacitor and the
total circuit voltage simultaneously. Draw where you should put
the oscilloscope leads on the circuit diagram shown above. Be sure to specify whether to use a
bottom ground or middle ground setup.
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v. Perform the measurement described in part d and record your observations (LABELS/UNITS):
vi. Explain in detail what is physically happening so that you see shark fins in the above circuit.
vii. What do you see when you increase the square wave frequency to higher and higher values?
What physically causes this?
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Lab 7 – Final comments before you start working on your own.
• Only charge electrolytic capacitors in the correct direction! Here let me show you…
Lab 7 – Procedure – On your own.
Experiment 2: Big capacitance = slow decay: DMM measurements of discharge only.
• Create an RC circuit with R=47kΩ and C=1000µF (electrolytic). Charge an electrolytic
capacitor without resistance (fast) in the correct direction and then switch to discharge through
a resistor (bypassing the voltage supply).
• Use a stopwatch to make voltage measurements across the capacitor as a function of time.
Collect more data initially as the capacitor discharges more rapidly. Collect data until you
reach ¼ of your initial starting voltage. This process should take a little more than one minute.
1. Construct a graph of the capacitor discharging: capacitor voltage vs. time.
2. Does your graph seem to behave as your theoretically predicted relationship using the
function solution to a differential equation (1 complete sentence explanation)?
GRAPH 1
3. Use your graph to find how long it takes to drop from peak voltage to half this value.
This is the half-life of the circuit. What is the half-life in seconds? Compare this to the
product of R and C. (Check your answers with me!)
1. Measured half life = ______________seconds
2. R*C = _____________ seconds
3. Mathematically, what should R*C represent?
ANS: The time is takes for the voltage to fall to ____ of its initial value.
4. Use your graph to find how long it takes to drop from one half-peak voltage to one
quarter peak voltage (another halving step). This is another half-life of the circuit and
should be about the same as in #2. Get me if the second half-life time is not about the
same as the first.
5. The fact that each halving of the decaying voltage occurs in the same amount of time is
a unique property of what mathematical function?
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6. Make a graph of ln(V(t)) vs. t. This should give you a straight line. The slope of this
line is -1/RC. Use this slope to calculate R*C. Get me if your value is not close to the
value of RC using the labeled values. GRAPH 2.
7. In the previous graph, I claimed that the slope was -1/RC. Why mathematically is this
true?
8. Imagine that you are a biologist modeling exponential population growth with
P(t) = P(0)e rt . If you had a database containing population versus time data, how
would you determine the population growth rate, r?
!
9. Why are seconds the units of R*C?
Experiment 3: Little capacitance (rapid charging and discharging measured on an oscilloscope).
1. Construct the same RC circuit as in experiment 1: R = 1 kΩ, C = 0.1 µF, square wave with
V MIN = 0 V & V MAX = 3 V and frequency of 1,000 Hz.
2. Measure the half-life of this rapidly decaying RC circuit. Be sure to stretch and maneuver your
oscilloscope screen to make an accurate measurement. This value should be in the
microsecond range or you are making an error. (Remember how to use SECONDS/DIV!!!)
T 1/2 reported = _____________ seconds.
3. Use your DMM to obtain and accurate value for R. Then use this value and the half-life to
calculate C, the capacitance of the tiny capacitor. Use the same math procedure we first
practiced on page 2, Theory Worksheet #1.
4. CLEAN UP LAB STATION.
C reported = _____________ Farads.
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Lab 7 – Report Guidelines
I will be paying careful attention to your solutions in the above pages. These solutions will
be worth approximately one third of your lab report points this week. (Ellipsis means you should not
need help here on what to write about).
1. Title - …
2. Goals – Something about differential equations and RC circuits.
3. Theory –
• What is the discharging differential equation? What physics principles did we use to
create this differential equation: name at least 3 (do not derive the equation or its
solution, we did that together already)?
• How did we solve the discharging differential equation and what was our solution?
What kind of functional decay do you find when discharging a capacitor?
4. Procedure – No more than 3 sentences for each experiment please!!!
5. Results
NO TYPING TODAY. YOUR RESULTS ARE YOUR TWO VERY NEATLY
MADE GRAPHS FOR THE LARGE DISCHARGING CAPACITOR (EXPERIMENT
2) AND ALL YOUR PAGES OF NOTES/SOLUTIONS.
6. PROBLEMS – YOU ALREADY DID THEM ON THESE HERE PAGES!!!
7. Conclusion –
a. Describe how an RC circuit works, and how you proved this experimentally by testing
with both a large RC and a small RC. (~ 2 sentences)
b. Explain how your lab today had you perform a complete physics research project (~3
sentences total):
1. Model some feature of the universe with a differential equation.
2. Solve the differential equation to find some particular behavior of the
universe.
3. Experimentally verify that your mathematical model correctly describes
this aspect of the universe (this is the part the Aristotle messed up).
c. As a last optional paragraph, discuss if you believe physics can tell us about reality or
merely models reality. (But not too long.) Note: I will keep my opinions to myself to
stay out of trouble!
d. As a last optional paragraph, explain how I might make this lab simpler in the future
without giving up academic content. Or do you like it the way it is? (Honest feedback
only, please.)
8. Attachments including all the notes you took on the pages of this handout.
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