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