phys 212 recitation review problems · points (e.g. “r = a”) or ... (labeled), a voltmeter and...

Physics 212 Spring 2013 Final Exam Recitation Review

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PHYS 212 Recitation Review Problems

The purpose of today’s recitation activity is to allow you to test yourself on the range of conceptual topics and

problem solving skills that you will see on the final exam. This should not be thought of as a sample exam – no

effort has been made to duplicate the number, style and content balance of the actual final (you have sample

exams for that purpose). Instead the problems here are designed to highlight some common issues that might

arise on the final.

We suggest that you try to solve these problems on your own and then, after all of the group members (working at

their own pace) have finished solving any individual problem, that you only then discuss that problem in your

group. Leave no group mate behind… (but feel free to work ahead if you finish before your group mates). There

are probably more problems here than you will have time to complete and discuss during the recitation. Solutions

will be posted Thursday night on Piazza and on the recitation pages (after the last recitation).

Exercise 1: Maxwell’s Equations Below are two very standard devices. Answer the following questions about how you would determine

the electric and magnetic fields as well as their direction at a point P. Also, determine at what radii, if

anywhere, the fields will be zero (or very nearly zero). Answers can be “nowhere,” or individual

points (e.g. “r = a”) or ranges (e.g. “a < r < 2a”). Note that this includes radii both inside and outside

the device, but in the plane pictured.

Parallel Plate Capacitor:

Starting with uncharged plates, a constant current has been and continues to flow as pictured

1) Which of Maxwell’s equations do you use to calculate the E field at P? _________________

2) In what direction is the electric field at P? _________________

3) In what direction is the magnetic field at P? _________________

P r

I

I

h

k

i

P r

I

j

i

Side View Top View

I

I


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Exercise 1: Maxwell’s Equations: Parallel Plate Capacitor continued…

4) In what direction is the Poynting vector at P? _________________

5) For what finite [finite is >0 <∞] radii r (if any) is the electric field zero? _________________

6) For what finite radii r (if any) is the magnetic field zero? _________________

Solenoid:

Starting from zero current, the current has been and continues to increase linearly in time in the direction

pictureda

7) Which of Maxwell’s equations do you use to calculate the E field at P? _________________

8) In what direction is the electric field at P? _________________

9) Which of Maxwell’s equations do you use to calculate the B field at P? _________________

10) In what direction is the magnetic field at P? _________________

11) For what finite radii r (if any) is the electric field zero? _________________

12) For what finite radii r (if any) is the magnetic field zero? _________________

h

a

N

I(t)

I(t)

Side View Top View

k

i

P r

I(t) j

i


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Exercise 2: Circuits Below are two correctly wired circuits, similar to those used in labs. In addition to the circuit elements

you can observe, there are also wires going to a DC power supply (labeled), a voltmeter and an ammeter

(not labeled, but you can figure out which is which, the wires are grouped).

Your job is to identify which, if any, of the below diagrams are representative of the voltage or current

you would measure in the given situation. For your answers use the labels A – D.

A B C D

If your power supply had been on for a long time, and then shut off (turned into a wire), then:

the voltmeter would read _______________ & the ammeter would read ____________________

Now a core is inserted into the coil. Describe the one main effect on the voltage plot:

_______________________________________________________________________________

Power Supply

From To


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Exercise 2: Circuits continued… Again, for your answers use the labels A – D.

A B C D

If your power supply had been off for a long time, and then turned on (to a constant voltage), then:

the voltmeter would read _______________ & the ammeter would read ____________________

Power Supply

From To


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Exercise 3: More Circuits

Below are three circuits consisting of a battery (EMF ), an inductor (L) and two identical resistors (R).

At time t=0 the switches are simultaneously closed in all three circuits. In each of the circuits calculate

the current through the battery both just after the switch is closed (t = 0+) and a long time after the

switch is closed (t = ∞).

At t = 0+, Ibattery =

_______________ _______________ _______________

At t = ∞, Ibattery =

_______________ _______________ _______________

Exercise 4: Charge in Hollow Conductor A very small, charged, metal sphere is placed

inside a thin conducting spherical shell of

radius B without touching it. Two Gaussian

spheres of radii A and C are used to find the net

flux inside and outside the shell

1)Suppose the two Gaussian surfaces have the same electric flux. What are the charges on the inner and

outer surfaces of the conducting shell?

Qinner = ____________________ Qouter = ____________________

2)Suppose the electric flux through the outer Gaussian surface is three times that through the inner

Gaussian surface. What are the charges on the inner and outer surfaces of the conducting shell?

Qinner = ____________________ Qouter = ____________________

3) If the electric flux through the outer Gaussian surface is zero, what is the electric flux through the

inner Gaussian surface?

Fluxinner = ____________________________________________________________


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Exercise 5: Magnetic Fields Electrically charged particles of equal mass and charge magnitude (though different sign) are moving

through regions of space in which there may be magnetic fields. In each case, shown is the sign of the

charge and a portion of the path the charge follows through the region. (These are top views looking

down on charges moving in the plane of the page.) All of the charges enter the regions with the same

initial velocity. There are no external electric fields and in answering the below questions you should

ignore any fields created by the moving charged particles themselves

In which region (A-F) is the component of the magnetic field out of the page largest? __________

In which region (A-F) is the component of the magnetic field into the page largest? __________

In which, if any, of the regions MUST the total B field be zero (A-F or none) ? __________


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Exercise 6: Random Forces These remaining problems ask you to determine the direction of forces on given objects. If the force is

zero, state that, otherwise give a direction (up, down, left, right, out of page, into page)

1) Consider two electric dipoles, each consisting of two equal and opposite point charges at the ends of

an insulating rod. The dipoles sit oriented as shown below.

In what direction will the force be (if any) on the dipole on the left? ___________________

2) Consider the iron filings diagram at left. The

pictured B field is generated by currents running

in the two coils of wire.

In what direction will the force be (if any) on

the upper coil?

__________________________________ ___________________

3) Consider the equipotential map (streaks parallel to equipotentials)

pictured at left, generated by three charges.

In what direction will the force be (if any) on the middle charge?

_______________________________________________


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Exercise 7: RL & RC Circuits

In experiments you have looked at both series RL & RC circuits, saw how to measure the time constants

in these circuits and how, from plots of current and voltage, to determine the inductance or capacitance

and the resistance in the circuit. In this problem we repeat part of that experiment (values are different

than those used in the lab). You set up a series RL or RC and drive it with a power supply, which feeds

in a square wave potential (on then off).

Using this setup, you record the voltage and current from the power supply:

From the information above determine the time constant of the circuit , the resistance R and

capacitance C or inductance L (tell us which!) that makes up the circuit. Note that although your

numbers will be approximate (you have to read them off of the plot) I always choose very nice numbers

for this type of problem. The space provided should be sufficient for all your calculations.

Circle one!

or R L C


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Exercise 8: Driven Series RLC

Consider an ideal series RLC circuit driven by a function generator. Voltmeters across the resistor,

capacitor and inductor record the following:

You also note that the power dissipated by the resistor oscillates, but peaks at Pmax = 16 mW.

(a) Identify each of the traces (i.e. name the component whose voltage corresponds to each trace)

Solid : ______________________

Dot: ______________________

Dash: ______________________

(b) Are we driving above or below the resonance frequency? How do you know?


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Exercise 8: Driven Series RLC continued…

In the remainder of this problem your goal is to read the plot and determine certain facts about the

circuit, including the values of the resistance, capacitance and inductance. In all parts you must first

determine an equation for the value sought, in terms of values that are given, that you can read off

the plot, or that you have previously determined. Then, and only then, should you plug in numerical

values. You will receive NO CREDIT if you do not first give an expression in terms of variables. Note

that in some cases you will need to make up your own variable names. Please try to keep to the

standards we used in class. Please box your answers.

(c) At what frequency f is the function generator driving the circuit?

(d) What is the resistance R of the resistor? HINT: Resistors dissipate power

(e) What is the amplitude of the oscillating current I0?


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DON’T FORGET: Equations with variables BEFORE plugging in numbers

(f) What is the value of the current at time t = 0? Why?

(g) What is the capacitance C of the capacitor?

(h) What is the inductance L of the inductor?


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DON’T FORGET: Equations with variables BEFORE plugging in numbers

(i) What is the amplitude of the power supply voltage 0,PSV ?

(j) What is the resonance frequency 0 of this circuit? Does this compare to your answer in part (c) as

you predicted in part (b)?


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Exercise 9: Transmission Line

The rest of this review is an extended question dealing with transmission lines. There are a variety of

transmission lines used in the world. A simple example is two wires running next to each other with

current flowing one direction in one and the opposite in the other. Another example that you considered

in mastering physics was the coaxial cable, where current flowed up the inside wire and back along the

outer shield.

In this problem you will calculate the properties of a microstrip transmission line. It consists of two thin

parallel plates of width w and length , separated by a small distance d (they are typically held apart by a

dielectric, but to make your life simple let’s just pretend there is air between the plates). It is shown

both in side view and front view below.

The dimensions are such that you should assume that any fields created by the transmission line are

confined to the region between the two plates.

We use transmission lines to carry power from batteries or power supplies to loads (typically modeled as

resistors):

In this problem you will calculate the capacitance and inductance of the microstrip transmission line and

then study energy flow at DC.

NOTE: PLEASE READ THIS CAREFULLY

In several parts of this problem you will be asked to calculate something that will require the use of one

of Maxwell’s equations. Make sure that you state the name of the equation and the write it in the form

that you plan to use it before you do that part. You do not need to describe the equation, but you do

need to be explicit in the calculations and draw and label anything that you need to use to do the

calculation. I will not provide any further drawings. Please duplicate drawings from this page

(simplified to remove the perspective of course) when you think they will be useful.

Do not forget to give both magnitude and direction of vector quantities.


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Exercise 9A: Capacitance of the Microstrip Transmission Line

In the first two parts of this problem (A and B) we will consider the transmission line in isolation (no

battery or load resistor).

Calculate the capacitance of the transmission line.


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Exercise 9B: Inductance of the Microstrip Transmission Line

Calculate the inductance of the transmission line.

NOTE: There are two ways to do this. If you don’t recall either of them then I suggest that you at least

send some current through the transmission line and calculate the magnetic energy between the plates.


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Exercise 9C: DC Power Transmission with the Microstrip Transmission Line

We now connect the transmission line to a battery (EMF ) on the left and a resistor (resistance R) on the

right, as pictured at the beginning of this problem. We are interested in what happens a long time after

this connection has been made (after any transient behavior has passed).

(a) What is the electric field between the plates? HINT: This is much easier than you probably think

now that the battery fixes the potential difference between the plates.

(b) What is the magnetic field between the plates? HINT: You probably already did this in B. Feel

free to just quote your previous result, but rewrite the current in terms of what we now know in this

problem.

(c) What is the Poynting vector between the plates?

(d) Integrate the Poynting vector over a relevant area and show that the result simplifies to what you

would expect given the meaning of the Poynting vector.


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Exercise 9D: Transients and AC Transmission in the Coaxial Cable

You calculated that the transmission line has an inductance per unit length and a capacitance per unit

length. A typical way to model the behavior of the transmission line is as a collection of inductors and

capacitors, as pictured below left, or even more simply as just a single inductor and capacitor, as

pictured below right.

These are “lossless” models – we are ignoring the resistance of the transmission line itself.

(a) Let’s first think about the transient behavior of this circuit. The instant after you attach a battery () on the left and a resistive load (R) on the right, what is the current through the load? Why? Describe

what the inductive and capacitive parts of the transmission are behaving like at this instant.

(b) A long time after the battery and load resistor R have been connected what is the current through

R? Why? Describe what the inductive and capacitive parts of the transmission line are behaving

like at this instant.


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Exercise 9D: Transients and AC Transmission continued

Now instead of attaching a battery on the left, let’s attach a function

generator, driving a voltage 0 sinV V t . On the right we still have a

resistive load, but to make life simpler let’s assume that it is a very large

resistor R.

(c) You have already discussed the very low frequency (DC) behavior of the

transmission line. As we turn up the frequency of the power supply,

QUALITATIVELY describe (no equations) what happens to the

voltage that the load sees. Why?

(d) I said above that you would want to assume that the load had a very large resistance R. Typically

when we say that something is very large, what we mean is that it is much larger than something

else. At non-zero frequencies, what should R be much larger than (give an equation here)? Why?

phys 212 recitation review problems · points (e.g. “r = a”) or ... (labeled), a voltmeter and...

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