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PHYS 152, Summer 2008
Problem Set 3
Due at 9:00pm on Monday, August 4, 2008
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A Few Bumps on the Road
Learning Goal: To learn to apply the microscopic theory of conduction. A gauge-12 wire has diameter centimeters and length meters. When the voltage volts is applied to the ends of the wire, the current is amperes. The concentration of free electrons in the wire is per cubic meter.
Part A Find the resistivity of the wire.
Hint A.1 How to approach the problem Hint not displayed
Hint A.2 Definition of resistivity Hint not displayed
Part A.3 What is the current density? Part not displayed
Part A.4 What is the electric field? Part not displayed
Express your answer in ohm-meters. Use two significant figures.
ANSWER: (^) = 2.75×10−
Part B Find the mean time between electron collisions in the wire.
Hint B.1 How to approach the problem Hint not displayed
Hint B.2 Definition of current density Hint not displayed
Hint B.3 Definition of resistivity Hint not displayed
Express your answer in seconds. Use two significant figures.
ANSWER: (^) = (^) 2.20×10−
A Stretchable Resistor
A wire of length and cross-sectional area has resistance.
Part A What will be the resistance of the wire if it is stretched to twice its original length? Assume that the density and resistivity of the material do not change when the wire is stretched.
Hint A.1 Formula for the resistance of a wire Hint not displayed
Part A.2 Find the cross-sectional area of the stretched wire Part not displayed
Express your answer in terms of the wire's original resistance.
ANSWER: =
Current and Current Density at a Junction
Consider the juncion of three wires as shown in the diagram.
The magnitudes of the current density and the diameters for wires 1 and 2 are given in the table. The current directions are indicated by the arrows.
Wire
Current density ( )
Diameter ( )
1 3.0 2. 2 5.0 3.
Part A Find the current in wire 3.
Hint A.1 How to approach the problem Hint not displayed
Hint A.2 Kirchhoff's rule Hint not displayed
copper is.
Part A Find the drift velocity of the electrons in the wire.
Part A.1 Find the current density first Part not displayed
Hint A.2 Current density and the drift speed Hint not displayed
Express your answer in meters per second, to two significant figures.
ANSWER: (^) = 3.30×10−
Note that this wire is carrying more current density than is carried by most household wiring in everyday use. With the given amount of current flowing, the cord would be hot to the touch if it were under a rug or had otherwise restricted air flow around it. It would certainly be considered unsafe by standard electrical safety codes.
Even though this wire is carrying a large amount of current for its size, the drift velocity of the electrons is tiny (less than one millimeter per second). This reflects the fact that there is a huge number of free (mobile) electrons in the wire. Let us illustrate this fact with a calculation.
Part B The population of the Earth is roughly six billion people. If all free electrons contained in this extension cord are evenly split among the humans, how many free electrons ( ) would each person get?
Part B.1 Find the volume first Part not displayed
Use two significant figures.
ANSWER: (^) = 7.50×10^13
These free electrons undergo frequent collisions with atoms, slowing down and generating heat. How many collisions occur in such a conductor? Let us find out.
Part C Find the total number of collisions ( ) that all free electrons in this extension cord undergo in one second.
Hint C.1 Consider a single electron How many collisions per second does each electron undergo per second?
Part C.2 Find the time between collisions Find the algebraic expression for the mean time between collisions.
Express your answer in terms of , the mass of the electron , the charge of the electron and the concentration of the electrons.
ANSWER:
ANSWER: =
1.80×10^37
' ote that does not depend on the applied electric field. The drift speed, however, does.
Electrical Safety
( ost of us have experienced an electrical shoc) one way or another in our lives. ( ost electrical shoc) s we receive are minor ones from wooly sweaters or from shoes. * owever, some shoc) s, especially from outlets or power mains, can be fatal. This + uestion will show you how to estimate the current through a human body when subject to an electrical shoc).
Imagine a situation in which a person accidentally touches an electrical soc) et with both hands. , y modeling the arm and the chest to be a cylindrical tube with a total length , cross-sectional area , and resistivity , you can calculate the
current in amperes through the person when a potential difference of is applied across the two hands. Assume that the current flows
only through the modeled cylindrical tube.
Part A What is the current flow through the body?
Hint A.1 Calculating the resistance Hint not displayed
Part A.2 Obtaining the expression for current Part not displayed
Express your answer numerically to two significant figures.
ANSWER: (^) = (^) 0.037 A
The following are the effects of current on humans-
1 mA = A or less- barely noticeable.
1 to / mA- strong surprise. / to 15 mA- unpleasant, victims able to detach from source. 15 to 0 5mA- painful, dangerous. 0 5 mA or more- fatal. These values vary according to sex, age, and weight.
The Low-Flow Pipe
Learning Goal: To learn to apply and combine the concepts of current, current density, and microscopic 1 hm2 s law.
An aluminum 3 12 gauge3 wire has a diameter of 0.205 centimeters. The resistivity of aluminum is ohm-meters. The electric
field in the wire changes with time as newtons per coulomb, where time is measured in seconds.
Part A Find the current through the conductor at time 5.0 seconds.
Hint A.1 How to approach the problem Hint not displayed
Part A.2 The equation for resistance
Note that the internal resistance is usually indicated as a separate resistor drawn next to the "battery" symbol. It is important to keep in mind that this resistor with resistance is actually inside the battery.
In all diagrams, stands for emf, for the internal resistance of the battery, and for the resistance of the external circuit. As usual, we'll
assume that the connecting wires have negligible resistance. We will also assume that both the ammeter and the voltmeter are ideal: That is, the ammeter has negligible resistance, and the voltmeter has a very large resistance.
Part A For the circuit shown in the diagram , which potential difference corresponds to the terminal voltage of the battery?
ANSWER: (^) between points K and L between points L and M between points K and M
Keep in mind that the "resistor" with resistance is actually inside the battery.
The next several questions refer to the four diagrams shown here labeled A, B, C, and D.
Part B In which diagram(s) (labeled A - D) does the ammeter correctly measure the current through the battery?
Hint B.1 How an ammeter works Hint not displayed
Enter the letter(s) of the correct diagram(s) in alphabetical order. For example if both A and C are correct enter AC.
ANSWER: (^) CD
Part C
In which diagram is the current through the battery nearly zero?
Hint C.1 How to approach the problem Hint not displayed
ANSWER: A
B
C
D
Diagram A is the only one in which the current through the battery is the same as the current through the voltmeter. Since the latter has a very large resistance, this current is essentially zero.
Part D
In which diagram or diagrams does the ammeter correctly measure the current through the resistor with resistance?
Hint D.1 How to approach the problem Hint not displayed
Enter the letter(s) of the correct diagram(s) in alphabetical order. For example if both A and C are correct enter AC.
ANSWER: (^) CD
Part E
In which diagram does the voltmeter correctly measure the terminal voltage of the battery? Choose the best answer.
Hint E.1 How a voltmeter works A voltmeter works by measuring the voltage of anything to which it is connected in parallel in the circuit. As a result, we would like the voltmeter to have a very high internal resistance so that not much current flows through it.
ANSWER: A
B
C
D
In diagrams A and B, the voltmeter readings would actually be quite close to the terminal voltage if the ammeter has a very low resistance, and the voltmeter, a very high one. However, diagram C clearly shows the best way to connect the voltmeter in order to measure the terminal voltage.
Part F
In which diagram does the voltmeter read almost zero?
Enter the letter(s) of the correct diagram(s) in alphabetical order. For example if both A and C are correct enter AC.
ANSWER: (^) D
The voltmeter in diagram D is connected to two points that are also connected by a wire that has, presumably, very low resistance. Therefore, the charge flowing through that wire will not lose an appreciable amount of potential energy, and the potential difference (voltage) is nearly zero.
Part G
Express your answer in volts. Use three significant figures.
ANSWER: (^) = (^) 10.
Since the ends of the resistor with resistance are attached to the terminals of the battery, the voltage across the resistor is the same as that between the terminals of the battery.
Part M How much work does the battery connected to the 21.0-ohm resistor perform in one minute?
Hint M.1 How to approach the problem Hint not displayed
Part M.2 Find the charge Part not displayed
Express your answer in joules. Use three significant figures.
ANSWER: (^) = (^) 360
Current in an Ionic Solution
Current passes through a solution of sodium chloride. In 1.00 second, ions arrive at the negative electrode and
ions arrive at the positive electrode.
Part A What is the current passing between the electrodes?
Hint A.1 General considerations on ionic solutions A conductor may contain various different kinds of moving charged particles with different charges, concentrations, and drift velocities. For example, in an ionic solution, current can be carried by both positive ions and negative ions; the total current is found by adding up the currents due to each kind of charged particle present in the solution.
Part A.2 Find the current due to the positive ions Find the current due to the positive sodium ions.
Hint A.2.a Definition of current Hint not displayed
Express your answer in milliamperes to three significant figures.
ANSWER: (^) = (^) 4.
Express your answer in milliamperes to three significant figures.
ANSWER: (^) 10.
Part B
What is the direction of the current?
Hint B.1 Conventional direction of current Hint not displayed
ANSWER: (^) away from the negative electrode toward the negative electrode
Two Resistors
Two resistors of resistances and , with , are connected to a voltage source with voltage. When the resistors are connected in
series, the current is. When the resistors are connected in parallel, the current from the source is equal to.
Part A Let be the ratio. Find.
Part A.1 Calculate the source voltage Part not displayed
Part A.2 Find another expresssion for the source voltage Part not displayed
Hint A.3 Equate the two expressions for voltage Hint not displayed
Hint A.4 Formula for the roots of a quadratic equation Hint not displayed
Hint A.5 General answer Hint not displayed
Round your answer to the nearest thousandth.
ANSWER: (^) 0.
Battery, Ammeter, and Resistors
An ammeter is connected in series to a battery of voltage and a resistor of unknown resistance. The ammeter reads a current. Next, a
resistor of unknown resistance is connected in series to the ammeter, and the
ammeter's reading drops to. Finally, a second resistor, also of resistance , is
connected in series as well. Now the ammeter reads.
Assuming that the dielectric is inserted at a constant rate, find the current as the slab is inserted.
Part A.1 What is the effect of the dielectric on capacitance? Part not displayed
Part A.2 What is the current in the circuit? Part not displayed
Part A.3 What is the initial capacitance? Part not displayed
Part A.4 What is the change in capacitance? Part not displayed
Express your answer in terms of any or all of the given variables , , , , , and , the permittivity of free space.
ANSWER:
Kirchhoff's Loop Rule Conceptual Question
The circuit shown belowconsists of four different resistors and a battery. 9 ou don't know the strength of the battery or the value any of the four resistances.
Part A Select the expressions that will be equal to the voltage of the battery in the circuit, where , for example, is the potential drop across resistor A.
Hint A.1 Kirchhoff's voltage rule for closed circuit loops Hint not displayed
Check all that apply.
ANSWER:
Resistance and Wire Length
9 ou have been given a long length of wire. 9 ou measure the resistance of the wire, and find it to be. 9 ou 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?
Part A.1 Find the resistance of the wire segments Part not displayed
Part A.2 Resistors in parallel Part not displayed
Express your answer in terms of and.
ANSWER:
Throw the Switch
Hint B.1 How to approach the problem Hint not displayed
Part B.2 Find the equivalent resistance of the circuit when the switch is closed Part not displayed
Part B.3 Find the voltage across bulb A when the switch is closed Part not displayed
Hint B.4 How to determine whether choice D is correct Hint not displayed
Part B.5 Find the voltage across bulb B when the switch is closed Part not displayed
Part B.6 Find the voltage across bulb B when the switch is open Part not displayed
Check all that apply.
ANSWER: (^) The potential difference across A increases. The potential difference across B doubles. The potential difference across B drops to zero. The potential difference across D is unchanged.
Resistance from Microscopic Ohm's law
Your task is to calculate the resistance of a simple cylindrical resistor with wires connected to the ends, such as the carbon composition resistors that are used on electronic circuit boards. Imagine that the resistor is made by squirting material whose conductivity is into a cylindrical mold
with length and cross-sectional area. Assume that this material satisfies Ohm's law. (It should if the resistor is operated within its power
dissipation limits.)
Part A What is the resistance of this resistor?
Hint A.1 General approach Hint not displayed
Part A.2 Microscopic Ohm's law Part not displayed
Part A.3 Find the voltage from the electric field Part not displayed
Part A.4 Find the current from the current density Part not displayed
Hint A.5 Ohm's law for the resistor Hint not displayed
Express the resistance in terms of variables given in the introduction. Do not use or in your answer.
ANSWER:
Real resistors vary tremendously in overall size. The larger the size, the more power the resistor can dissipate without heating to the point that it is dangerous to nearby components or that the material of which it is constructed begins to change its conductivity (i.e., so that the resistance would no longer be constant). The amount of resistance is determined by the conductivity of the material of the resistor, which can vary over more than 20 orders of magnitude. Commercially available resistors vary from 0.1 ohm or less to more than ohm.
Batteries in Series or Parallel
You are given two circuits with two batteries of emf and internal resistance each. Circuit A has the batteries connected in series with a
resistor of resistance , and circuit B has the batteries connected in parallel to an equivalent resistor.
Note that the symbol should be entered in your answers as EMF.
Part A
ANSWER: = 0.064 W
Part E For what ratio of and would power dissipated by the resistor of resistance be the same for circuit A and circuit B?
Hint E.1 Getting started Hint not displayed
Hint E.2 Finding
Hint not displayed
ANSWER: = 1
Part F Under which of the following conditions would power dissipated by the resistance in circuit A be bigger than that of circuit B?
Hint F.1 How to think about the problem This part is very similar to the previous part with the equality sign replaced by less than or greater than signs.
Some answer choices overlap; choose the most restrictive answer.
ANSWER:
Summary 17 of 17 items complete (99.77% avg. score) 74.81 of 75 points, 10 of 10 extra credit
