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DC Circuits: Ch 32
• Voltage – Starts out at highest point at “+” end of
battery
• Voltage drops across lightbulbs and other
sources of resistance.
• Voltage increases again at battery.
I
+
Voltage highest
Voltage zero
The following circuit uses a 1.5 V battery and
has a 15 W lightbulb.
a. Calculate the current in the circuit (P = IV)
b. Calculate the voltage drop across the lightbulb.
c. Sketch a graph of voltage vs. path (battery, top
wire, resistor, bottom wire)
Resistors in Series
• Same current (I) passes through all resistors
(bulbs)
• All bulbs are equally bright (energy loss, not
current loss)
• Voltage drop across each resistor (V1,V2, V3)
V = V1 + V2 + V3
V = IR1 + IR2 + IR3
V = I(R1 + R2 + R3)
Req = R1 + R2 + R3
V = IReq
Resistors in Parallel
• Current splits at the junction
• Same Voltage across all resistors
I = I1 + I2 + I3
I1 = V
R1
I=V
Req
1 = 1
Req R1
+
1
R2
+
1
R3
Which combination of auto headlights will produce
the brightest bulbs? Assume all bulbs have a
resistance of R.
For the Bulbs in Series:
Req = R + R = 2R
For the Bulbs in Parallel
1 = 1
+
1
Req R
R
1 = 2
Req R
Req = R/2
The bulbs in parallel have less resistance and
will be brighter
What current flows through each resistor in the
following circuit? (R = 100 W)
Req = R1 + R2
Req = 200 W
V = IReq
I = V/Req
I = 24.0 V/ 200 W = 0.120 A
Calculate the current through this circuit, and the
voltage drop across each resistor.
Req = 400 W + 290 W
Req = 690 W
V = IR
I = V/Req
I = 12.0 V/690 W
I = 0.0174 A
Vab = (0.0174A)(400W)
Vab = 6.96 V
Vbc = (0.0174A)(290W)
Vbc = 5.04 V
What current flows through each of the
resistors in this circuit? (R = 100 W)
1 = 1
+ 1
Req 100 W 100 W
1/Req = 2/100 W
Req = 50 W
I = V/Req = 24.0 V/50 W = 0.48 A
DC Circuits: Ex 4
What current will flow through the circuit shown?
1 = 1
+
Rp = 500 W
Rp = 290 W
1
700 W
Req = 400 W + 290 W
Req = 690 W
V = IR
I = V/R
I = 12.0 V/690 W
I = 0.017 A or 17 mA
Example 4
Calculate the equivalent resistance in the following
circuit.
DC Circuits: Ex 5
What current is flowing through just the 500 W
resistor?
First we find the voltage drop across the first
resistor:
V = IR = (0.017 A)(400 W)
V = 6.8 V
The voltage through the resistors in parallel will be:
12.0 V – 6.8 V = 5.2 V
To find the current across the 500 W resistor:
V = IR
I = V/R
I = 5.2 V/500 W = 0.010 A = 10 mA
DC Circuits: Ex 6
Which bulb will be the brightest in this
arrangement (most current)?
Bulb C (current gets split running through A and
B)
What happens when the switch is opened?
– C and B will have the same brightness (I is constant
in a series circuit)
DC Circuits: Ex 5
What resistance would be present between points A
and B?
(ANS: 41/15 R)
EMF and Terminal Voltage
• Batteries - source of emf (Electromotive Force),
E (battery rating)
• All batteries have some internal resistance r
Vab = E – Ir
Vab = terminal(useful)voltage
E = battery rating
r = internal resistance
EMF: Example 1
A 12-V battery has an internal resistance of 0.1 W.
If 10 Amps flow from the battery, what is the
terminal voltage?
Vab = E – Ir
Vab = 12 V – (10 A)(0.10 W)
Vab = 11 V
EMF: Example 2
Calculate the current in the following circuit.
1/Req = 1/8 W + ¼ W
Req = 2.7 W
Req = 6 W + 2.7 W
Req = 8.7 W
1/Req = 1/10 W + 1/8.7 W
Req = 4.8 W
Everything is now in series
Req = 4.8 W + 5.0 W + 0.50 W
Req = 10.3 W
V = IR
I = V/R
I = 9.0 V/10.3 W
I = 0.87 A
EMF: Example 2a
Now calculate the terminal(useful)voltage.
V = E – Ir
V = 9.0 V – (0.87 A)(0.50 W)
V = 8.6 V
Grounded
•
•
•
•
Wire is run to the ground
Houses have a ground wire at main circuit box
Does not affect circuit behavior normally
Provides path for electricity to flow in
emergency
Kirchoff’s Rules
1. Junction Rule - The sum of the
currents entering a junction must
equal the sum of currents leaving
2. Loop Rule - The sum of the changes
in potential around any closed path =
0
Kirchoff Conventions
The “loop
current” is not a
current.
Just a direction
that you follow
around the loop.
Kirchoff Conventions
Kirchoff’s Rule Ex 1
Junction Rule
I 1 = I 2 + I3
Loop Rule
Main Loop
6V – (I1)(4W) – (I3)(9W) = 0
Side Loop
(-I2)(5W) + (I3)(9W) = 0
I1 = I2 + I3
6V – (I1)(4W) – (I3)(9W) = 0
(-I2)(5W) + (I3)(9W) = 0
Solve Eqn 1
(-I2)(5W) + (I3)(9W) = 0
(I3)(9W) = (I2)(5W)
I3 = 5/9 I2
Eqn 3
Eqn 2
Eqn 3
Substitution into Eqn 2
6V – (I2 + I3)(4W) – (I3)(9W) = 0
6 – 4I2 -4I3 - 9I3 = 0
6 – 4I2 - 13I3 = 0
I3 = 5/9 I2 (from last slide)
6 – 4I2 - 13(5/9 I2) = 0
6 = 101/9 I2
I2 = 0.53 A
I3 = 5/9 I2 = 0.29 A
I1 = I2 + I3 = 0.53 A + 0.29 A = 0.82 A
Kirchoff’s Rule Ex 2
I1 = I2 + I3
Loop Rule
Main Loop
9V – (I3)(10W) – (I1)(5W) = 0
Side Loop
(-I2)(5W) + (I3)(10W) = 0
(-I2)(5W) + (I3)(10W) = 0
(I2)(5W) = (I3)(10W)
I2 = 2I3
9V – (I3)(10W) – (I1)(5W) = 0
9V – (I3)(10W) – (I2 + I3)(5W) = 0
9 –10I3 – 5I2 – 5I3 = 0
9 –15I3 – 5I2 = 0
9 –15I3 – 5(2I3) = 0
9 –25I3 = 0
I3 = 9/25 = 0.36 A
I3 = 9/25 = 0.36 A
I2 = 2I3 = 2(0.36 A) = 0.72 A
I1 = I2 + I3 = 0.36A + 072 A = 1.08 A
Kirchoff’s Rules: Ex 3
Calculate the currents in
the following circuit.
I1 + I 2 = I 3
Bottom Loop (clockwise)
10V – (6W)I1 – (2W)I3 = 0
Top Loop (clockwise)
-14V +(6W)I1 – 10 V -(4W)I2 = 0
Work with Bottom Loop
10V – (6W)I1 – (2W)I3 = 0
I1 + I 2 = I 3
10 – 6I1 – 2(I1 + I2) = 0
10 – 6I1 – 2I1 - 2I2 = 0
10 - 8I1 - 2I2 = 0
10 = 8I1 + 2I2
5 = 4I1 + I2
I2 = 5 - 4I1
Working with Top Loop
-14V +(6W)I1 – 10 V -(4W)I2 = 0
24 = 6I1 - 4I2
12 = 3I1 - 2I2
12 = 3I1 - 2(5 - 4I1)
22 = 11I1
I1 = 22/11 = 2.0 Amps
I2 = 5 - 4I1
I2 = 5 – 4(2) = -3.0 Amps
I1 + I 2 = I 3
I3 = -1.0 A
Batteries in Series
• If + to -, voltages add (top
drawing)
• If + to +, voltages subtract
(middle drawing = 8V, used
to charge the 12V battery as
in a car engine)
Batteries in Parallel
• Provide large current when
needed
Extra Kirchoff Problems
I1 = -0.864 A
I2 = 2.6 A
I3 = 1.73 A
A Strange Example
Calculate I (-0.5 Amps (we picked wrong
direction))
a. Calculate the equivalent resistance. (2.26 W)
b. Calculate the current in the upper and lower
wires. (3.98 A)
c. Calculate I1, I2, and I3 (0.60 A, 0.225 A, 1.13 A)
d. Sketch a graph showing the voltage through the
circuit starting at the battery.
RC Circuits
•
•
•
•
Capacitors store energy (flash in a camera)
Resistors control how fast that energy is released
Car lights that dim after you shut them
Camera flashes
DVc + DVr = 0
Q - IR = 0
C
(Divide by R)
Q - I =0
RC
(I = -dQ/dt)
Q + dQ = 0
RC dt
 = time constant (time to reach 63% of full
voltage)
 = RC
A versatile relationship
V = Vo e-t/RC
I = Io e-t/RC
Q = Qo e-t/RC
I generally find voltage, then use V=IR and Q=VC
RC Circuits: Ex 1
What is the time constant for an RC circuit of
resistance 200 kW and capacitance of 3.0 mF?
 = (200,000 W)(3.0 X 10-6 F)
 = 0.60 s
(lower resistance will cause the capacitor to charge
more quickly)
RC Circuits: Ex 2
What will happen to the bulb (resistor) in the
circuit below when the switch is closed (like a
car door)?
Answer: Bulb will glow brightly initially, then dim
as capacitor nears full charge.
RC Circuits: Ex 3
An uncharged RC circuit has a 12 V battery, a 5.0
mF capacitor and a 800 kW resistor. Calculate
the time constant.
 = RC
 = (5.0 X 10-6 F)(800,000 W)
 = 4.0 s
What is the maximum charge on the capacitor?
Q = CV
Q = (5.0 X 10-6 F)(12 V)
Q = 6 X 10-5 F or 60 mF
What is the voltage and charge on the capacitor
after 1 time constant?
V = (0.632)(12 V) = 7.584 V
Q = (0.632)(60 mF) = 38 mF
Consider the circuit below. Calculate:
a. The time constant (6 ms)
b. Maximum charge on the capacitor (3.6 mC)
c. Time to reach 99% of maximum charge (28 ms)
d. Current when charge = ½ Qmax (300 mA)
e. Maximum current (600 mA)
f. The charge when the current is 20% of the
maximum value. (2.9 mC)
Discharging the RC Circuit
V = Vo e-t/RC
RC Circuits: Ex 4
An RC circuit has a charged capacitor C = 35 mF
and a resistance of 120W. How much time will
elapse until the voltage falls to 10 percent of its
original (maximum) value?
V = Vo e-t/RC
0.10Vo =Voe-t/RC
0.10 =e-t/RC
ln(0.10) = ln(e-t/RC)
-2.3 = -t/RC
2.3 = t/RC
t = 2.3RC
t = (2.3)(120W)(35 X 10-6 F)
t = 0.0097 s or 9.7 ms
RC Circuits: Ex 5
If a capacitor is discharged in an RC circuit, how
many time constants will it take the voltage to
drop to ¼ its maximum value?
V = Vo e-t/RC
0.25Vo =Voe-t/RC
0.25 =e-t/RC
ln(0.25) = ln(e-t/RC)
-1.39 = -t/RC
1.39 = t/RC
t = 1.39RC
A fully charged 1.02 mF capacitor is in a circuit
with a 20.0 V battery and a resistor. When
discharged, the current is observed to
decrease to 50% of it’s initial value in 40 ms.
a. Calculate the charge on the capacitor at t=0
(20.4 mC)
b. Calculate the resistance R (57 W)
c. Calculate the charge at t = 60 ms (7.3 mC)
The capacitor in the drawing has been fully
charged. The switch is quickly moved to
position b (camera flash).
a. Calculate the initial charge on the capacitor. (9
mC)
b. Calculate the charge on the capacitor after 5.0
ms. (5.5 mC)
c. Calculate the voltage after 5.0 ms (5.5 V
d. Calculate the current through the resistor after
5.0 ms (0.55 A)
Meters
• Galvanometer
– Can only handle a small current
• Full-scale Current Sensitivity (Im)
– Maximum deflection
• Ex:
– Multimeter
– Car speedometer
Measuring I and V
Measuring Current
– Anmeter is placed in series
– Current is constant in series
Measuring Voltage
– Voltmeter placed in parallel
– Voltage constant in parallel circuits
– Measuring voltage drop across a resistor
Anmeter (Series)
Voltmeter (parallel)
DC Anmeter
• Uses “Shunt” (parallel) resistor
• Shunt resistor has low resistance
• Most of current flows through shunt, only a little
through Galvanometer
• IRR = IGr
Meters: Ex 1
What size shunt resistor should be used if a
galvanometer has a full-scale sensitivity of 50 mA
and a resistance of r= 30 W? You want the meter
to read a 1.0 A current.
Voltage same through both (V=IR)
I R R = I gr
Since most of the current goes through the shunt
IR ~ 1 A
IRR = Igr
(1 A)(R) = (50 X 10-6 A)(30 W)
R = 1.5 X 10-3 W or 1.5 m W
Meters: Ex 2
Design an anmeter that can test a 12 A vacuum
cleaner if the galvanometer has an internal
resistance of 50 W and a full scale deflection of 1
mA.
IRR = Igr
(12 A)(R) = (1 X 10-3 A)(50 W)
R = 4.2 X 10-3 W or 4.2 m W
DC Voltmeter
• Resistor in series
• Large R for resistor (keeps current low in
Galvanometer)
• V = I(R + r)
Meters: Ex 3
What resistor should be used in a voltmeter that
can read a maximum of 15 V? The galvanometer
has an internal resistance of 30W and a full scale
deflection of 50 mA.
V = I(R + r)
12 V = (50 X 10-6 A)(R + 30W)
R + 30W = 12 V
50 X 10-6 A
R + 30W = 300,000 W
R ~ 300,000 W
Meters: Ex 4
Design a voltmeter for a 120 V appliance with and
internal galvanometer resistance of 50 W and a
current sensitivity of 1 mA.
(ANS: R = 120,000 W)
Electric Power
• Watt
• 1 Watt = 1 Joule
1 second
P = I2R
“Twinkle, twinkle, little star. Power equals I2R”
Power: Ex 1
Calculate the resistance of a 40-W auto headlight
that operates at 12 V.
P = I2R
V = IR (so I =V/R)
P = V2R
R2
P = V2
R
P = V2
R
R = V2 = (12 V)2 = 3.6 W
P
40 W
Household Electricity
• Kilowatt-hour
• You do not pay for power, you pay for energy
1 kWh = (1000 J)(3600 s) = 3.60 X 106 J
1s
Power: Ex 2
An electric heater draws 15.0 A on a 120-V line.
How much power does it use?
P = I2R
V = IR so R = V/I
P = I2V
I
P = IV = (15 A)(120V) = 1800 W or 1.8 kW
Power: Ex 3
How much does it cost to run it for 30 days if it
operates 3.0 h per day and the electric company
charges 10.5 cents per kWh?
Hours = 30 days X 3.0 h/day = 90 h
Cost = (1.80 kW)(90 h)($0.105/kWh) = $17
Power: Ex 4
A lightening bolt can transfer 109 J of energy at a
potential difference of 5 X 107V over 0.20 s.
What is the charge transferred?
V = PE/Q
Q = E/V = 109 J/ 5 X 107V = 20 C
What is the current?
I = DQ/Dt
I = 20 C/0.2 s = 100 A
What is the power?
P = I2R
V = IR so R = V/I
P = I2V
I
P = IV = (100 A)(5 X 107V ) = 5 X 109 W
Household Electricity
• Circuit breakers – prevent “overloading” (too
much current per wire)
• Metal melts or bimetallic strip expands
Household Electricity:
Ex 1
Determine the total current
drawn by all of the
appliances shown.
P = IV
I = P/V
Ilight = 100W/120 V = 0.8 A
Ilight = 100W/120 V = 0.8 A
Iheater = 1800W/120 V = 15 A
Istereo = 350W/120 V = 2.9 A
Ihair = 1200W/120 V = 10.0 A
Itotal = 0.8A + 15.0A + 2.9A + 10.0A = 28.7 A
This would blow the 20 A fuse
DC vs AC
DC
AC
•Electrons flow constantly
•Electrons flow in short burst
•Electrons flow in only one
direction
•Electrons switch directions
(60 times a second)
•Batteries
•House current
DC vs AC
http://www.ibiblio.org/obp/electricCircuits/AC/AC_1.html
Jump Starting a Car
POSITIVE TO POSITIVE
(or your battery could explode)