PHYS-1500 PHYSICAL MODELING
Class 22a: Capacitors
FALL 2006
NAME _________________________________
A capacitor is a device that stores charge. It can be viewed as two parallel plates placed close
together. One plate carries positive charge and the other an equal amount of negative charge. The
capacitor can be charged by connecting it to a voltage source, one plate to the high potential side
and the other to the low. Then, the amount of charge that is deposited on the capacitor is
proportional to the voltage applied. The constant of proportionality is called the capacitance, C.
If the voltage difference between the two plates is V, and the magnitude of the charge on each
plate is q, then q = CV. Conversely, if we wish to know the voltage across a charged capacitor, it
is V = q/C.
In this exercise, you will examine how capacitors react when connected in series with a voltage
source and a resistor. The sketch shows the situation.
The source provides a voltage E, and it must equal the sum of the voltage across the resistor, iR,
and that across the capacitor. Then, E = iR + q/C. This equation is modeled in the spreadsheet
that has been provided. The spreadsheet calculates changes in q as q = it, and, from the
equation above, i = (E/R) – q/(RC). E can be either D.C., A.C., or a mixture of both, since it is
given by E = E 1 + E 2 sin 0t. The quantity RC, which appears in the equation for i, is called the
time constant of the circuit.
1. Consider q0 = 0, E1 = 10 V, E2 = 0, C = 1.0 ×10-6 F, and R = 1.0 ×106 .
a) What is the time constant of the circuit?
RC =__________________
units
b) Use t = 0.05 s. Sketch the graph of q vs. t. What is the value of q at t = RC?
q =__________________
units
1
2. Set R = 2.0 ×106 , but leave everything else the same.
a) What is the time constant of the circuit?
RC =__________________
units
b) Use t = 0.05 s. Sketch the graph of q vs. t. What is the value of q at t = RC?
q =__________________
units
3. Set R = 0.50 ×106 , but leave everything else the same.
a) What is the time constant of the circuit?
RC =__________________
units
b) Use t = 0.05 s. Sketch the graph of q vs. t. What is the value of q at t = RC?
q =__________________
units
4. Set R = 1.0 ×106 , set E1 = 0, and q0 = 1.0 ×10-5 C, but leave everything else the same.
a) What is the time constant of the circuit?
RC =__________________
units
b) Use t = 0.05 s. Sketch the graph of q vs. t. What is the value of q at t = RC? What is the
relationship between the value of q found in this section and that found in part 1?
q =__________________
units
2
5. Set q0 = 0, E1 = 0, E2 = 10 V, C = 1.0 ×10-6 F, and R = 1.0 ×106 . Find i(max), the maximum
value of i, for 0 = 0.05, 0.2, 1.0, 10, and 100 rad/s. Then do the same for C = 2.0 ×10-6 F.
Enter the results in the table shown, and graph i(max) vs. 0 for both values of C on the same
graph.
t
1.5
0.5
0.1
0.01
0.001
o
i (max)
C 1.0 ×10-6
2.0 ×10-6
0.05
0.2
1.0
10
100
Sketch the graph below.
6. Is an RC circuit a better conductor of current at high frequency or low frequency? (Circle the
correct answer.) HIGH FREQUENCY
LOW FREQUENCY
Which capacitor is the better conductor of current, the larger or the smaller? (Circle the
correct answer.)
LARGER
SMALLER
3
Class 22b: Inductors
An inductor is a device that stores energy in a magnetic field. It can be viewed as a closely
wound coil of wire. When the inductor carries a current, it produces a magnetic field within the
coil. The inductor can be charged by connecting it to a voltage source, with a resistor in series,
one end to the high potential side and the other to the low. Then, the voltage difference between
the two ends of the inductor is proportional to the rate of change of the current through the
inductor. The constant of proportionality is called the inductance, L. Then, the voltage difference
between the two ends is V = L di/dt.
In this exercise, you will examine how inductors react when connected in series with a voltage
source and a resistor. The sketch shows the situation.
The source provides a voltage E, and it must equal the sum of the voltage across the resistor, iR,
and that across the inductor. Then, E = iR + L(di/dt). This equation is modeled in the spreadsheet
that has been provided. The spreadsheet calculates changes in i as i = (di/dt)t, and, from the
equation above, di/dt = (E/L) – q/(L/R). E can be either D.C., A.C., or a mixture of both, since it
is given by E = E 1 + E 2 sin 0t. The quantity L/R, which appears in the equation for di/dt, is
called the time constant of the circuit.
1. Consider i0 = 0, E1 = 10 V, E2 = 0, L = 1.0 ×10-1 H, and R = 1.0 ×101 .
a) What is the time constant of the circuit?
L/R =__________________
units
b) Use t = 0.0005 s. Sketch the graph of i vs. t. What is the value of i at t = L/R?
i =__________________
units
4
2. Set L = 2.0 ×10-1 H, but leave everything else the same.
a) What is the time constant of the circuit?
L/R =__________________
units
b) Use t = 0.0005 s. Sketch the graph of i vs. t. What is the value of i at t = L/R?
i =__________________
units
3. Set L = 0.5 ×10-1 H, but leave everything else the same.
a) What is the time constant of the circuit?
L/R =__________________
units
b) Use t = 0.0005 s. Sketch the graph of i vs. t. What is the value of i at t = L/R?
i =__________________
units
4. Set L = 1.0 ×10-1 H, set E1 = 0, and i0 = 1.0 A, but leave everything else the same.
a) What is the time constant of the circuit?
L/R =__________________
units
b) Use t = 0.0005 s. Sketch the graph of i vs. t. What is the value of i at t = L/R? What is
the relationship between the value of i found in this section and that found in part 1?
i =__________________
units
5
5. Set i0 = 0, E1 = 0, E2 = 10 V, L = 1.0 ×10-1 H, and R = 1.0 ×101 . Find i(max), the maximum
value of i, for 0 = 10, 100, 500, and 1000 rad/s. Then do the same for L = 2.0 ×10-1 H. Enter
the results in the table shown, and graph i(max) vs. 0 for both values of L on the same graph.
t
0.01
0.005
0.0005
0.0002
o
i (max)
L 1.0 ×10-1
2.0 ×10-1
10
100
500
1000
Sketch the graph below.
6. Is an LR circuit a better conductor of current at high frequency or low frequency? (Circle the
correct answer.) HIGH FREQUENCY
LOW FREQUENCY
Which inductor is the better conductor of current, the larger or the smaller? (Circle the correct
answer.)
LARGER
SMALLER
6