Assignment 12: DC Circuits
Due: 8:00am on Friday, February 17, 2012
Note: To understand how points are awarded, read your instructor's Grading Policy.
Equivalent Resistance
Consider the network of four resistors shown in the diagram, where = 2.00 , = 5.00 , = 1.00 , and
= 7.00 . The resistors are connected to a constant voltage of magnitude .
Part A
Find the equivalent resistance of the resistor network.
Hint A.1 How to reduce the network of resistors
Hint not displayed
Hint A.2 Find the resistance equivalent to and
Hint not displayed
Hint A.3 Three resistors in series
Hint not displayed
Express your answer in ohms.
ANSWER:
=
9.43
Correct
Part B
Two resistors of resistance = 3.00 and = 3.00 are added to the network, and an additional resistor of
resistance = 3.00 is connected by a switch, as shown in the diagram..
Find the equivalent resistance of the new resistor network when the switch is open.
Hint B.1 How to reduce the extended network of resistors
Hint not displayed
Hint B.2 Find the resistance equivalent to , , and
Hint not displayed
Hint B.3 Four resistors in series
Hint not displayed
Express your answer in ohms.
ANSWER:
=
13.5
Correct
Part C
Find the equivalent resistance of the resistor network described in Part B when the switch is closed.
Hint C.1 How to reduce the network of resistors when the switch is closed
Hint not displayed
Hint C.2 Find the resistance equivalent to and
Hint not displayed
Hint C.3 Four resistors in series
Hint not displayed
Express your answer in ohms.
ANSWER:
=
8.60
Correct
Series Resistors with Different Areas
Four wires are made of the same highly resistive material, cut to the same length, and connected in series.
1. Wire 1 has resistance and cross-sectional area .
2. Wire 2 has resistance and cross-sectional area .
3. Wire 3 has resistance and cross-sectional area .
4. Wire 4 has resistance and cross-sectional area .
A voltage is applied across the series, as shown in the figure.
Part A
Find the voltage across wire 2.
Hint A.1 Resistance of wires
Hint not displayed
Hint A.2 Find the current through wire 2
Hint not displayed
Hint A.3 The voltage across wire 2
Hint not displayed
Give your answer in terms of , the voltage of the battery.
ANSWER:
=
Correct
Brightness of Light Bulbs Ranking Task
Part A
Consider a circuit containing five identical light bulbs and an ideal battery. Assume that the resistance of each
light bulb remains constant. Rank the bulbs (A through E) based on their brightness.
Hint A.1 How to approach the problem
Hint not displayed
Hint A.2 Comparing bulb A to bulb B
Hint not displayed
Hint A.3 Comparing bulb D to bulb E
Hint not displayed
Hint A.4 Comparing bulb C to bulb D or E
Hint not displayed
Hint A.5 Comparing bulb C to bulb A or B
Hint not displayed
Rank from brightest to dimmest. To rank items as equivalent, overlap them.
ANSWER:
Correct
Now consider what happens when a switch in the circuit is opened.
Part B
What happens to the brightness of bulb A?
Hint B.1 How to approach this part
Hint not displayed
Hint B.2 Consider changes in resistance
Hint not displayed
ANSWER:
It gets dimmer.
It gets brighter.
There is no change.
Correct
Part C
What happens to bulb C?
Hint C.1 How to approach this part
Hint not displayed
Hint C.2 Find the current in bulb C earlier
Hint not displayed
Hint C.3 Find the current in bulb C now
Hint not displayed
ANSWER:
It gets dimmer.
It gets brighter.
There is no change.
Correct
This is why appliances in your home are always connected in parallel. Otherwise, turning some of them on or off
would cause the current in others to change, which could damage them.
Resistance and Wire Length
You have been given a long length of wire. You measure the resistance of the wire, and find it to be . You then
cut the wire into identical pieces .
Part A
If you connect the pieces in parallel as shown , what is the total
resistance of the wires connected in parallel?
Hint A.1 Find the resistance of the wire segments
Hint not displayed
Hint A.2 Resistors in parallel
Hint not displayed
Express your answer in terms of and .
ANSWER:
=
Correct
Kirchhoff's Rules and Applying Them
Learning Goal: To understand the origins of both of Kirchhoff's rules and how to use them to solve a circuit
problem.
This problem introduces Kirchhoff's two rules for circuits:
Kirchhoff's loop rule: The sum of the voltage changes across the circuit elements forming any closed loop
is zero.
Kirchhoff's junction rule: The algebraic sum of the currents into (or out of) any junction in the circuit is
zero.
The figure shows a circuit that illustrates the concept of loops, which are
colored red and labeled loop 1 and loop 2. Loop 1 is the loop around the entire circuit, whereas loop 2 is the
smaller loop on the right. To apply the loop rule you would add the voltage changes of all circuit elements around
the chosen loop. The figure contains two junctions (where three or more wires meet)--they are at the ends of the
resistor labeled . The battery supplies a constant voltage , and the resistors are labeled with their resistances.
The ammeters are ideal meters that read and respectively.
The direction of each loop and the direction of each current arrow that you draw on your own circuits are arbitrary.
Just assign voltage drops consistently and sum both voltage drops and currents algebraically and you will get
correct equations. If the actual current is in the opposite direction from your current arrow, your answer for that
current will be negative. The direction of any loop is even less imporant: The equation obtained from a
counterclockwise loop is the same as that from a clockwise loop except for a negative sign in front of every term
(i.e., an inconsequential change in overall sign of the equation because it equals zero).
Part A
The junction rule describes the conservation of which quantity? Note that this rule applies only to circuits that are
in a steady state.
Hint A.1 At the junction
Hint not displayed
ANSWER:
current
voltage
resistance
Correct
Part B
Apply the junction rule to the junction labeled with the number 1 (at the bottom of the resistor of resistance ).
Hint B.1 Elements in series
Hint not displayed
Answer in terms of given quantities, together with the meter readings and and the current .
ANSWER:
Correct
If you apply the juncion rule to the junction above , you should find that the ezpression you get is equivalent to
what you just obtained for the junction labeled 1. Obviously the conservation of charge or current flow enforces
the same relationship among the currents when they separate as when they recombine.
Part C
Apply the loop rule to loop 2 (the smaller loop on the right). Sum the voltage changes across each circuit element
around this loop going in the direction of the arrow. Remember that the current meter is ideal.
Hint C.1 Elements in series have same current
Hint not displayed
Hint C.2 Sign of voltage across resistors
Hint not displayed
Hint C.3 Voltage drop across ammeter
Hint not displayed
Express the voltage drops in terms of , , , the given resistances, and any other given quantities.
ANSWER:
Correct
Part D
Now apply the loop rule to loop 1 (the larger loop spanning the entire circuit). Sum the voltage changes across
each circuit element around this loop going in the direction of the arrow.
Express the voltage drops in terms of , , , the given resistances, and any other given quantities.
ANSWER:
Correct
There is one more loop in this circuit, the inner loop through the battery, both ammeters, and resistors and . If
you apply Kirchhoff's loop rule to this additional loop, you will generate an extra equation that is redundant with
the other two. In general, you can get enough equations to solve a circuit by either
1. selecting all of the internal loops (loops with no circuit elements inside the loop) or
2. using a number of loops (not necessarily internal) equal to the number of internal loops, with the extra
proviso that at least one loop pass through each circuit element.
Exercise 26.28
Part A
In the circuit shown in the figure , find the magnitude of current in the
upper branch.
ANSWER:
=
0.800
Correct
Part B
Find the magnitude of current in the middle branch.
ANSWER:
=
0.200
Correct
Part C
Find the magnitude of current in the lower branch.
ANSWER:
=
0.600
Correct
Part D
What is the potential difference of point relative to point ?
ANSWER:
=
-3.20
Correct
Finding Current by Changing Resistors
A battery provides a voltage of 7.00 and has unknown internal resistance . When the battery is connected
across a resistor of resistance = 6.00 , the current in the circuit is = 1.00 .
Part A
If the external resistance is then changed to = 4.00 , what is the value of the current in the circuit?
Hint A.1 How to approach the problem
Hint not displayed
Hint A.2 Internal resistance explained
Hint not displayed
Hint A.3 Find the internal resistance
Hint not displayed
Express your answer numerically in amperes.
ANSWER:
=
1.40
Correct
Problem 26.59
The two identical light bulbs in Example 26.2 in the textbook (Section 26.1) are connected in parallel to a different
source, one with and internal resistance 0.7 . Each light bulb has a resistance (assumed
independent of the current through the bulb).
Part A
Find the current through each bulb.
Express your answer using two significant figures. Enter your answers numerically separated by a comma.
ANSWER:
, =
2.4,2.4
Correct
Part B
Find the potential difference across each bulb.
Express your answer using two significant figures. Enter your answers numerically separated by a comma.
ANSWER:
, =
4.7,4.7
Correct
Part C
Find the power delivered to each bulb.
Express your answer using two significant figures. Enter your answers numerically separated by a comma.
ANSWER:
, =
11,11
Correct
Part D
Suppose one of the bulbs burns out, so that its filament breaks and current no longer flows through it. Find the
power delivered to the remaining bulb.
Express your answer using two significant figures.
ANSWER:
=
18
Correct
Part E
Does the remaining bulb glow more or less brightly after the other bulb burns out than before?
ANSWER:
more brightly
less brightly
Correct
Exercise 26.38
A galvanometer having a resistance of 20.0 has a 1.00 shunt resistance installed to convert it to an ammeter.
It is then used to measure the current in a circuit consisting of a 15.0 resistor connected across the terminals of
a 25.0- battery having no appreciable internal resistance.
Part A
What current does the ammeter measure?
ANSWER:
1.57
Correct
Part B
What should be the true current in the circuit (that is, the current without the ammeter present)?
ANSWER:
1.67
Correct
Part C
By what percentage is the ammeter reading in error from the true current?
Express your answer using two significant figures.
ANSWER:
= 6.0
Correct %
Problem 26.79
Point in the figure is maintained at a constant potential of 400 above
ground.
Part A
What is the reading of a voltmeter with the proper range and with resistance when connected between
point and ground?
ANSWER:
=
114
Correct
Part B
What is the reading of a voltmeter with resistance ?
ANSWER:
=
263
Correct
Part C
What is the reading of a voltmeter with infinite resistance?
ANSWER:
=
267
Correct