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AP Physics 10: Current Electricity
A.
Circuit Elements
1. purpose of circuit is to control the direction and
amount of charge that flows to circuit elements
2. electric current, I = Q/t (A)
a. ampere, A = C/s
b. current flows from a high voltage to low voltage
1. Vhigh  Vlow (conventional current)
2. analogous to water running down hill
c. electron flow through conductor
1. I = Nqe/t
a. N is number of electrons
b. qe is charge per electron (1.6 x 10-19 C)
2. Vlow  Vhigh (opposite conventional current)
d. Ohm’s law: V = IR (I = V/R)—river analogy
1. electron flow through metal is analogous to
water running down a river channel
2. river current depends on channel steepness
(steeper channel = faster current): I  V
3. river current depends on channel roughness
(rougher channel = slower current): I  1/R
4. Ohm's law is a generalization, not natural law
3. electromotive force, emf (E or V)
a. general term for any devise that can generate a
voltage gradient, E = Vhigh – Vlow
b. E produced by chemical reaction (battery) or
induction (generator)
c. terminal voltage (V) and emf (E) of a battery
1. current flow out of a battery is limited by the
internal resistance, r, of the battery
2. terminal voltage, V = E ± Ir
3. (–) when discharging, (+) when charging
4. resistor, R ()
a. ohm,  = V/A
b. general term for anything that resists current (light
bulb, heater, motor)
c. R = L/A
A
L
depends on the material and state
a. resistivity,  (•m)
b. increases as temperature increases
2. proportional to length, L (m)
3. inversely proportional to area, A (m2)
d. as current flows through a resistor
1. I is constant, I = V/R
2. V decreases, V = -IR
3. power is consumed, P = IV
Steps
Algebra
P = W/t
start with
substitute QV for W
P = (QV)/t
P = (Q/t)V
regroup
P = IV
substitute I for Q/t
substitute IR for V
P = I(IR) = I2R
P = (V/R)2R = V2/R
substitute V/R for I
5. capacitor (discussed earlier)
a. Q = CV measured in Coulombs, C
b. C =єoA/d measured in Farads, F
(єo = 8.85 x 10-12 C2/N•m2)
c. UC = ½QV = ½CV2 = ½Q2/V measured in Joules, J
d. E = V/d measured in V/m or N/C
1.
Name __________________________
B.
6.
symbols for circuit elements
7.
Circuit diagram of a battery, resistor, capacitor and
switch
Series and Parallel Circuit Design
1. batteries in series
a. total voltage: Vs = V1 ± V2 ± V3
b. add when + to – (current exits + pole)
c. subtract when + to + (current exits higher V)
2. resistors
a. series
R1
R2
R3
1. Rs = R1 + R2 + R3
Steps
Algebra
start with
R = Ltot/A
substitute L1 + L2 + L3 for Ltot
Rs = (L1 + L2 + L3)/A
distribute
Rs = L1/A + L2/A + L3/A
substitute R for L/A
Rs = R1 + R2 + R3
2. current, I, is the same for all resistors
3. share of voltage drop, Vn = IRn
b. parallel
R1
R2
R3
1. 1/Rp = 1/R1 + 1/R2 + 1/R3
Steps
Algebra
start with
R = L/Atot
substitute A1 + A2 + A3 for Atot Rp = L/(A1 + A2 + A3)
inverse both sides
1/Rp = (A1 + A2 + A3)/L
distribute
1/Rp = A1/L + A2/L + A3/L
substitute 1/R for A/L
1/Rp = 1/R1 + 1/R2 + 1/R3
2. voltage, V, is the same for all resistors
3. share of current, In = V/Rn
3.
capacitors
a. series
C1
C2
C3
1/Cs = 1/C1 + 1/C2 + 1/C3
Steps
Algebra
start with
C =єoA/dtot
substitute d1 + d2 + d3 for dtot Cs = єoA/(d1 + d2 + d3)
inverse both sides
1/Cs = (d1 + d2 + d3)/єoA
distribute
1/Cs = d1/єoA + d2/єoA + d3/єoA
substitute 1/C for d/єoA
1/Cs = 1/C1 + 1/C2 + 1/C3
2. charge, Q, is the same for all capacitors
3. share of voltage drop, Vn = Q/C
b. parallel
C1
1.
C2
1.
C3
Cp = C1 + C2 + C3
Steps
Algebra
start with
C =єoAtot/d
substitute A1 + A2 + A3 for Atot Cp = єo(A1 + A2 + A3)/d
distribute
Cp = єoA1/d + єoA2/d + єoA3/d
substitute C for єoA/d
Cp = C1 + C2 + C3
2. voltage, V, is the same for all capacitors
3. share of charge, Qn = CnV
4. Kirchhoff’s rules
a. loop rule: V = 0 for any complete loop
b. junction rule: Iin = Iout for any junction
5. determining the overall resistance, Rtot, problems.

look for small groups of series or parallel resistors
6.







o combine 8  + 4 : 1/Rp = 1/8 + 1/4  R = 2.7 
o combine 6  + 2.7  = 8.7 
o combine 8.7  + 10 : 1/Rp = 1/8.7 + 1/10  R = 4.8 
o combine 4.8  + 5.0  + 0.50  = 7.3 
Solving complex circuit problems
Determine overall voltage, Vtot
Determine overall resistance, Rtot
Determine current from the battery, I = Vtot/Rtot
Determine voltage loss in resistors in series with the
battery, V = ItotR
Determine voltage loss in parallel circuit elements
using Kirchhoff's loop rule
Determine current, I = V/R, and power, P = IV, for all
resistors
Determine charge, Q = CV, and energy, Uc = ½QV, for
all capacitors
Experiments
1.
Batteries and Bulbs Lab
a. Attach wires between one set of battery terminals and
one set of light bulb terminals so that the light bulb
lights. Diagram the circuit.
b.
(1) Connect a second light bulb to the circuit using
only one additional wire. Diagram the circuit
(2) The brightness of the first bulb (increased or
decreased).
(3) When one light bulb is unscrewed from its socket,
the other bulb (stays on or goes out).
(4) Why does the other light bulb go out? (This
arrangement of light bulbs is called series)
c.
(1) Connect the two bulbs with the battery using two
long and two short wires so that the electricity
from the battery can go through one bulb without
having to go through the other. Diagram the
circuit.
(2) The brightness of the bulbs in this arrangement
compared to the series circuit arrangement is
(brighter or dimmer).
(3) When one light bulb is unscrewed from its socket,
the other bulb (stays on or goes out).
(4) Why does the other light bulb remain lit? (This
arrangement of light bulbs is called parallel)
d.
2.
Series and Parallel Circuit Lab
How to use the Multimeter
Voltage: Move the red lead to the V/ slot, and then
turn the dial to 2-V DC (long line over three dashed lines).
Attach the red lead to the resistor/capacitor spring closest
to the + pole of the battery and the black lead to the spring
on the other side of the resistor/capacitor, then record the
measurement. This arrangement is called parallel. If the
reading begins with zeros, you can turn the dial to a more
sensitive level. Be sure to record the potential with the
correct power of 10.
Current: Move the red lead to the mA/A slot, and
then turn the dial to 1-A DC. Free the resistor wire nearest
the + pole of the battery from its spring (leave the other
wire in its spring). Attach the red lead to the spring nearest
the + pole of the battery. Pinch the black lead and free
resistor wire together with your finger and thumb. This
arrangement is called series. If the reading begins with
zeros, you can turn the dial to a more sensitive level. Be
sure to record the current with the correct power of 10.
Capacitance: Connect a 100,000- resistor (brownblack-yellow) and capacitor in series (Spring—Resistor—
Spring—Capacitor—Spring). Set the multimeter to 2 V DC
and connect the meter to the springs on either side of the
capacitor (not the resistor). Connect a wire from the (–)
terminal of the battery to the capacitor spring farthest from
the resistor. When ready to time, connect a wire from the
(+) terminal of the battery to the resistor spring farthest
from the capacitor. Record the time it takes for the voltage
to go from 0.200 V to 1.200 V. Capacitance is proportional
to time (Q = It = CV  C = (I/V)t = kt).
Part 1—Resistors
a. Measure the current and voltage for each resistor
listed below (each resistors is identified by its color
bands). Calculate the resistance.
Current
Voltage Resistance
Color Bands
(V)
(I)
(R = V/I)
Brown-Black-Brown
Orange-Orange-Brown
Green-Blue-Brown
b. (1) Determine the theoretical resistance when the
three resistors are attached in series.
(2) Place the three resistors in series, measure current
and voltage, and then calculate the total resistance.
Voltage (V)
Current (I)
Resistance (R = V/I)
(1) Connect two batteries and 1 bulb together in a
series circuit using two long and one short wire.
Diagram the circuit.
(3) Calculate the percent difference between the
measured and theoretical values for resistance.
c.
e.
(2) The additional battery makes the bulb (brighter or
dimmer).
(1) Connect the two batteries and three bulbs so that
each bulb shines independently and brightly.
Diagram the circuit.
(1) Determine the theoretical resistance when the
three resistors are attached in parallel.
(2) Place the three resistors in parallel, measure
current and voltage, and then calculate the total
resistance.
Voltage (V)
Current (I)
Resistance (R = V/I)
(3) Calculate the percent difference between the
measured and theoretical values for resistance.
(2) The bulbs are connected in (series or parallel).
(3) The batteries are connected in (series or parallel).
Part 2—Capacitors
a. Follow the instructions above to attach the 100 F
capacitor and 100,000  resistor. Record the time it
takes to go from 0.200 V to 1.200 V. Repeat with the
330F capacitor. (Discharge the capacitor by touching
a wire between the two sides of the capacitor.)
Capacitance,
Time, t (s)
C (F)
Trial 1
Trial 2
Average
100 F
2.
You double the voltage across a certain conductor and you
observe the current increases three times. What can you
conclude?
(A) Ohm's law is obeyed since the current still increases
when V increases
(B) Ohm's law is not obeyed
(C) This has nothing to do with Ohm's law
3.
Two wires, A and B, are made of the same material and
have equal lengths, but the resistance of wire A is four
times the resistance of wire B. How do their diameters
compare?
(A) DA = 4DB
(B) DA = 2DB
(C) DA = ½DB
(D) DA = ¼DB
4.
A wire of resistance R is stretched uniformly (keeping its
volume constant) until it is twice its original length. What
happens to the resistance?
(A) ¼R
(B) ½R
(C) 2R
(D) 4R
5.
When you rotate the knob of a light dimmer, what is being
changed in the electric circuit?
(A) power (B) current (C) voltage (D) both P and I
6.
Two light bulbs operate at 120 V, but one has a power
rating of 25 W while the other has a power rating of 100 W.
Which one has the greater resistance?
(A) 25 W bulb (B) 120 W bulb (C) tie
7.
Two space heaters are operated at 120 V. Heater A has
twice the resistance as heater B. Which one will give off
more heat?
(A) A
(B) B
(C) tie
(3) Calculate the total capacitance based on the
charging time (C = kt) , use the average k from (b).
8.
Which current flows from high to low voltage, electron flow
or conventional current?
(4) Calculate the percent difference between the
measured and theoretical values for capacitance.
9.
How many electrons (qe = -1.60 x 10-19 C) pass when a
current of 10 A runs for 25 minutes?
330 F
b. Calculate k (k = C/t) for each capacitor and the average.
c.
(1) Determine the theoretical capacitance when the
two capacitors are attached in series.
(2) Place the two capacitors in series and measure the
time it takes to charge the capacitors.
Trial 1
Trial 2
Average
(3) Calculate the total capacitance based on the
charging time (C = kt), use the average k from (b).
(4) Calculate the percent difference between the
measured and theoretical values for capacitance.
d
(1) Determine the theoretical capacitance when the
two capacitors are attached in parallel.
(2) Place the two capacitors in parallel and measure the
time it takes to charge the capacitors.
Trial 1
Trial 2
Average
Practice Problems
A. Circuit Elements
Questions 1-7 Briefly explain your answer.
1. Which is the correct way to light the light bulb with the
battery?
10. What is the current through a 200- resistor if the voltage
between its terminals is 15 V?
11. What is the internal resistance of a battery where E = 1.5 V
and the terminal voltage = 1.35 V when the current is 3 A?
12. In general, how is resistivity affected by changes in
temperature?
13. a.
Determine the electrical resistance of tungsten wire
( = 5.0 x 10-8 •m, L = 20 m and A = 1.0 x 10-6 m2).
b.
Determine the electrical resistance in the same piece
of tungsten after it is stretched to a length of 60 m.
14. What is the resistance in a 48- light that operates at 12 V?
15. What can increase the capacitance for a parallel plate
capacitor?
16. A parallel plate capacitor consists of two metal plates
separated by 0.006 m and is connected to a 100-V source.
The area of each plate is 0.04 m2. Determine the
a. capacitance.
b.
energy stored.
d.
electric field.
17. Show how a battery, bulb and two wires must be arranged
in order for the bulb to light. Use circuit element symbols.
18. A starter motor draws a current of 50 A through a cable for
5 s. Determine the number of
a. coulombs of charge which pass through the cable.
electrons that pass through the cable.
19. A 50-m long wire with cross section of 3 x 10-6 m2 has a
resistance of 0.5 . What is the resistivity?
20. What is the resistance in a light bulb that draws 500 mA
current at 3 V.
21. A 10  resistor is connected to a 120 V line. Determine
a. the current through the resistor.
b.
(D) both P and I
25. What is the voltage across the 4  resistor?
(A) 1/3V
(B) 1/2 V
(C) 2/3 V
(D) V
Questions 26-27 Current, I, enters a parallel circuit containing a
2  resistor and a 4  resistor.
26. What is the same for both resistors?
(A) P
(B) I
(C) V
(D) both P and I
charge on each plate.
c.
b.
24. What is the same for both resistors?
(A) P
(B) I
(C) V
the power dissipated in the resistor.
27. What is the current through the 4  resistor?
(A) 1/3 I
(B) ½ I
(C) 2/3 I
(D) I
Questions 28-29 Two light bulbs (resistors) are in series, with a
wire and switch connected parallel to one of the bulbs. How
does the brightness of the bulb when the switch is open
compare to its brightness when the switch is closed?
A
B
28. Bulb A?
(A) dimmer (B) same
(C) brighter (D) no light
29. Bulb B?
(A) dimmer (B) same
(C) brighter (D) no light
Questions 30-31 Two light bulbs are in series with a 12-V battery.
30. What happens to the total brightness when one light bulb is
removed?
(A) dimmer (B) same
(C) brighter (D) no light
31. What happens to the total brightness when one light bulb is
replaced by a wire?
(A) dimmer (B) same
(C) brighter (D) no light
Questions 32-33 Two light bulbs are parallel with a 12-V battery.
32. What happens to the total brightness when one light bulb is
removed?
(A) dimmer (B) same
(C) brighter (D) no light
33. What happens to the total brightness when one light bulb is
replaced by a wire?
(A) dimmer (B) same
(C) brighter (D) no light
22. What is the power rating of a theater light in which a
current of 10 A is caused by 120 V?
Questions 34-36 The three light bulbs have the same resistance.
C
A
B
B. Series and Parallel Circuit Design
Questions 23-41 Briefly explain your answer.
23. A 9-V battery is connected to three identical resistors in
series. What is the voltage across each resistor?
(A) 3 V
(B) 9 V
(C) 18 V
(D) 27 V
34. The current through bulb C compared to bulb A is
(A) ¼
(B) ½
(C) 2
(D) 4
Questions 24-25 A battery of voltage V is connected to a 4 
and 2  resistor in series.
35. The voltage drop across bulb C compared to bulb A is
(A) ¼
(B) ½
(C) 2
(D) 4
36. How much brighter is bulb C compared to A?
(A) ¼
(B) ½
(C) 2
(D) 4
I for 3 
I for 6 
Questions 37-39 Identical light bulbs are arranged in the circuit.
A
C
D
B
E
37. What is the correct order from greatest (1) to least (5)
current?
(A)
(B)
(C)
(D)
P for 6 
IA
IB
IC
ID
IE
1
2
4
2
1
2
4
2
3
4
2
1
3
4
2
1
3
1
1
4
38. Which generates the most light?
(A) A + B
(B) C + D (C) E
P for 3 
(D) C + D + E
39. Which has the smallest voltage drop?
(A) A
(B) C
(C) E
40. Two capacitors are in series with a 12-V battery. What
happens to the total capacitance when one capacitor is
replaced by a wire?
(A) decrease
(B) same
(C) increase
41. Two capacitors are in parallel a 12-V battery. What
happens to the total capacitance when one capacitor is
replaced by a wire?
(A) decrease
(B) same
(C) increase
42. Highlight the correct option for the following sentences.
a. (Current or Voltage) is the same for resistor in series.
b. (Current or Voltage) is the same for resistors in parallel.
c. You can disconnect one device without stopping the
current in a (series or parallel) circuit.
d. Two light bulbs arranged in (series or parallel) will
generate the most light.
43. 3  and 6  resistors in series connected to 6 V.
45. Given R1 = 6 , R2 = 12 , and R3 = 2  are arranged in
the following circuit. Determine the
R1
R3
R2
12 V
a.
total resistance.
b.
total current leaving the battery.
c.
voltage drop across R3.
d.
voltage, current and power for each resistor.
V
P
I
R1
R2
R3
e.
46. Highlight the correct option for the following sentences.
a. Capacitor (Charge or Voltage) is the same in series.
b. Capacitor (Charge or Voltage) is the same in parallel.
47. 3 F and 6 F capacitors in series connected to 6 V.
Ctot
Qtot
Total R
UC-tot
Total I
V3
Total P
V6
V for 3 
UC-3
V for 6 
UC-6
P for 3 
48. 3 F and 6 F capacitors in parallel connected to 6 V.
Ctot
P for 6 
44. 3  and 6  resistors in parallel connected to 6 V.
Show that the power dissipated in the resistors equals
the power generated by the battery?
Qtot
Total R
UC-tot
Total I
Q3
Total P
Q6
b. Determine current, voltage and power for each resistor.
V
P
Resistor
I
UC-3
1
UC-6
49. Three capacitors, C1 = 10 F, C2 = 20 F, C3 = 30 F, are
arranged as shown below. Determine the
C1
C3
120 V
5
12 
6
C2
55. Consider the following circuit
⁄switch
15 
a.
total capacitance.
b.
total charge stored on C3.
c.
voltage across C3.
d.
voltage, charge and potential energy for each capacitor.
V
Q
UC
50V
a.
6F
(15 F)
10 
When the switch is open, determine
(1) charge on the 6 F capacitor.
(2) energy stored in the 6 F capacitor.
C1
b.
C2
C3
When the switch is closed, determine
(1) Voltage across the capacitor.
(2) charge on the 6 F capacitor.
50. Show where a voltmeter (V) and ammeter (A) would be
placed in order to measure volts and amps in the resistor.
(3) energy stored in the 6 F capacitor.
(A/V)
(A/V)
51. Three 12 resistors can be connected in four different
ways. Determine overall resistance of each combination?
52. Determine the equivalent resistance when three resistors
rated at 2 , 4 , and 6  are connected in
series
Practice Multiple Choice
Briefly explain why the answer is correct in the space provided.
Questions 1-2 The four resistors have the lengths, L, and
cross-sectional areas, A, indicated and are made of
material with the same resistivity.
(A) L = 1 m, A = 1 m2
(B) L = 2 m, A = 1 m2
(C) L = 1 m, A = 2 m2
(D) L = 2 m, A = 2 m2
1. Which resistor has the least resistance?
2.
Which has the greatest resistance?
3.
An electric circuit contains a variable resistor connected to
a battery. Which graph best represents the relationship
between current and resistance in this circuit?
(A)
(B)
(C)
(D)
4.
A circuit consists of a 10- resistor, a 15- resistor, and a
20- resistor connected in parallel across a 9-V battery.
What is the equivalent resistance of this circuit?
(A) 0.2 
(B) 2 
(C) 4.6 
(D) 45 
parallel
53. A 100-W, 120-V lamp bulb is connected in parallel with a
60-W, 120-V lamp bulb. What is their combined
resistance?
54. Consider the following circuit.
5
a.
Rtot
Itot
90V
1
Determine the
6
12 
5.
Two 4.0- resistors are connected to a I6-V battery.
The power generated in the circuit is
(A) 8 W
(B) 16 W
(C) 32 W
(D) 64 W
Questions 6-7 relate to the four incomplete circuits below
composed of resistors R, all of equal resistance, and
capacitors C, all of equal capacitance. A battery that can
be used to complete any of the circuits is available.
R
R
C
R
(A)
(B)
Questions 10-12 refer to the circuit shown below.
10. The equivalent capacitance for this network is
(A) 10/7 F (B) 3/2 F
(C) 7/3 F
(D) 7 F
11. The charge stored in the 5-F capacitor is
(A) 360 C (B) 500 C (C) 710 C (D) 1,100 C
12. The electrical energy stored in the 5-F capacitor is
(A) 0.025 J (B) 0.050 J (C) 2.5 J
(D) 500 J
d
.
C
(C)
R
(D)
R
R
Questions 13-14 refer to partial electric circuit.
R
6.
Into which circuit should the battery be connected to obtain
the greatest steady power dissipation?
7.
Which circuit will retain stored energy if the battery is
connected to it and then disconnected?
13. The electrical resistance between point X and point Y is
(A) 4/3 
(B) 2 
(C) 11/4 
(D) 4 
8.
A battery with an internal resistance of 4  connected to a
l6- and a 20- resistor in series. The current in the 20-
resistor is 0.3 A.
14. The current is
(A) the same everywhere in the circuit
(B) greater at point X than at point Y
(C) greater in the 1  resistor than in the 2  resistor
(D) greater in the 2  resistor than in the 3  resistor
15. A 10- heater is used to heat water. If the heater draws 3
A for 100 s, how much energy is transferred to the water?
(A) 30
(B) 300
(C) 3,000 (D) 9,000
What is the emf, E, of the battery?
(A) 1.2 V
(B) 6.0 V
(C) 10.8 V
9.
(D) 12 V
Questions 16-18 The circuit consisting of four resistors and a
12-V battery.
A lamp, a voltmeter V, an ammeter A, and a battery with
zero internal resistance are connected as shown.
16. What is the current measured by the ammeter?
(A) 0.5 A
(B) 2 A
(C) 72 A
(D) 4 A
How would the addition of a second lamp affect the
ammeter and voltmeter readings?
A
V
A
V
(A) increase same
(B) decrease decrease
(C) same
increase (D) decrease decrease
17. What is the equivalent resistance of this circuit?
(A) 72 
(B) 3 
(C) 18 
(D) 0.33 
18. How much power is dissipated in the 36- resistor?
(A) 110 W (B) 3 W
(C) 48 W
(D) 4 W
19. What is the voltage between points X and Y?
Practice Free Response
1.
Consider the following circuit.
C1
C2

R3
R4
R2
R1
R5
9V
(A) 1 V
(B) 2 V
(C) 3 V
R6
C3
(D) 4 V
a.
20. What is the value of r?
Determine the following values.
R3 + R4
R3-4 + R5
Rtot
Itot
(A) 0 
(B) 1 
(C) 5 
(D) 20 
V5
I5
Questions 21-22 refer to the circuit shown below.
I3
C1 + C2
Q1+2
V1
V2
Q3
21. What is the current I1?
(A) 0.8 mA (B) 1.0 mA (C) 2.0 mA (D) 3.0 mA
22. How do the currents I1, I2, and I3 compare?
(A) I1 > I2 > I3
(B) I1 > I3 > I2
(C) I2 > I1 > I3
(D) I3 > I1 > I2
23. What percentage of the power generated by a 0.5 A, 120 V
electric motor is used to lift a 9 kg mass against gravity at
an average velocity of 0.5 m/s?
(A) 7%
(B) 13%
(C) 25%
(D) 75%
24. A wire of length L and radius r has a resistance R. What is
the resistance of a second wire made from the same
material that has a length ½L and a radius ½r?
(A) 4R
(B) 2R
(C) R
(D) ½R
b. Complete the table for each resistor and capacitor.
R
V
P
Resistor
I
Overall
R1
1
R2
3
R3
3
R4
3
R5
3
R6
3
Capacitor
C
C1
30 F
C2
15 F
C3
20 F
Q
V
Uc