# A2L Item 247

Goal: Link energy with electrical quantities

Source: 283 – energy in capacitor

Consider the following circuit. The capacitor is uncharged when switch S
is closed at t = 0. After current stops flowing and the capacitor is
fully charged the energy stored in the capacitor is:

1. V^2^/R
2. CE^2^/2
3. QV/2
4. V/RC
5. none of the above

### Commentary:

(3) The intent of this question is to provide students the opportunity
to distinguish a correct but uncommon form for the stored energy from a
number of other familiar forms.

# A2L Item 246

Goal: reason and link electrical quantities

Source: 283-Del C, E, U, etc. – cstV

Two
parallel conducting plates form a capacitor. With a metal cylinder of
length half the plate separation inserted between the plates, it is
connected to a battery with potential ΔV. The cylinder is now removed.
Which of the quantities C, ΔV, Q, E, and U change?

1. C, E, and U only
2. Q and U only
3. C and U only
4. E and U only
5. Q, ΔV, and U only
6. ΔV and E only
7. C and Q only
8. None of the above

### Commentary:

(8) Since ΔV does not change, E must because the distance between
plates doubles. If E changes, so must Q. If Q changes, so must C.
Finally, that U chages can be seen in a variety of ways.

There are many good follow up questions, such as: How would the
quantities change if the cylinder was made of a dielectric material?

# A2L Item 245

Goal: Reason and link electrical quantities.

Source: 283-Del C, E, U, etc – cstQ

Two parallel conducting plates form a capacitor. It is isolated and a
charge Q is placed on it. A metal cylinder of length half the plate
separation is then inserted between the plates. Which of the quantities
C, ΔV, Q, E, and U change?

1. C, E, and U only
2. ΔV and U only
3. C and U only
4. E and U only
5. C, ΔV, and U only
6. ΔV and E only
7. C and ΔV only
8. None of the above

### Commentary:

(8) Students who are formula bound find this a difficult question.
Obviously Q does not change. Depending on how students interpret the
question, they may conclude that E does or does not change. The value of
E in regions outside the cylinder does not change [Students taking this
interpretation may respond #5.], but inside the cylinder it is now zero.
Since E is now zero for half of the distance between the original
plates, both ΔV and U must change. That C also changes can be
appreciated in many different ways.

There are many good follow up questions, such as: Does it make a
difference where the cylinder is placed? How would the quantities change
if the cylinder was made of a dielectric material? Suppose a half
cylinder of length 2d were placed between the plates. How would
quantities change?

# A2L Item 244

Goal: Hone skill at using Lenz’s Law

Source: 283-CTQsas36

Consider the four situations below in which a wire loop lies in the
plane of a long wire. In which case(s) is the induced current in the
loop in the counterclockwise direction? [Note: if no velocity is
indicated, the loop is stationary.]

1. A only
2. B only
3. C only
4. D only
5. A and B
6. C and D
7. All cases.
8. None of the above.

### Commentary:

1. Discussing each of the cases is valuable. Students have difficulty
recognizing that the induced current is in a direction that causes the
field of the induced current to compensate for the change in flux. Even
when they nominally understand that, they have trouble reliably applying
the right hand rule.

# A2L Item 243

Goal: Hone the concept of line integral

Source: 283-CTQsas35

Three wires, each carrying the same current, I, are in a region of
space, as shown below. What could be the result of computing the left
hand side of Ampere’s law, , for the
three Amperian Loops shown?

1. Loop 1: μ0I, Loop 2: 2μ0I, Loop 3: 3μ0I
2. Loop 1: -μ0I, Loop 2: 0, Loop 3: μ0I
3. Loop 1: μ0I, Loop 2: 2μ0I, Loop 3: μ0I
4. Cannot be determined

### Commentary:

(4) The direction to integrate around the loop is not specified. The
only choice of responses that is possibly true is #2 and this would
require a clockwise integral around loop 1.

# A2L Item 242

Goal: Reason regarding circuits

Source: 283 Compare dissipated power with inductors

Consider the following circuits. Two identical batteries are
connected to two identical inductors in series with different resistors.
The switch is closed at t=0. Which is true regarding the energy
supplied by the battery to establish current I?

1. The battery in A does more work.
2. The battery in B does more work.
3. Both batteries do the same work.
4. Cannot be determined

### Commentary:

(4) It is not determined that either circuit can achieve the current I.
If I is to be interpreted as the ‘final’ current, then the answer is #2.
The lower the resistance the higher the final current. There is more
energy stored in the inductor if the final current is higher. In
addition, the energy dissipated in the resistor goes as i2R which is
larger for circuit B for every current larger than V/(2R).

# A2L Item 241

Goal: Reason regarding circuits

Source: 283 compare dissipated energy

Consider the following circuits. Two identical batteries are connected
to two identical capacitors in series with different resistors. The
capacitors are initially uncharged. Which statement is true regarding
the energy supplied by the batteries to charge the capacitor?

1. The battery in A does more work.
2. The battery in B does more work.
3. Both batteries do the same work.
4. Cannot be determined

### Commentary:

(3) The energy dissipated in the resistor is independent of the
resistance. Consider a time when the capacitor contains some charge Q.
If an additional charge dq is added, the battery does work dqV and the
increment of stored energy in the capacitor is (Q/C)dq. By conservation
of energy, the difference must have been dissipated in the resistor. The
difference, [V-(Q/C)]dq is independent of resistance.

# A2L Item 240

Goal: Reason regarding power in a circuit

Source: 283 circuit powers

Consider the circuit below. Which resistor has the greatest power
consumption?

1. The 50Ω resistor
2. The 10Ω resistor
3. The 1000Ω resistor
4. The 50Ω and 10Ω resistors
5. The 50Ω and 1000Ω resistors
6. The 10Ω and 1000Ω resistors
7. All have the same power.

### Commentary:

(1) The potential drop over the 10Ω and 1000Ω resistors is the same.
Since power goes as V^2^/R, more power is consumed in the 10Ω resistor
than the 1000Ω resistor. Further, since power also goes as I^2^R and only
a fraction of the current through the 50Ω resistor flows through the 10Ω
resistor, the 50Ω resistor must dissipate the most energy.

# A2L Item 239

Goal: Reason with Lenz’s Law

Source: 238-730 Lenz’s Law 2

For which of the following situations will the current flow clockwise?
<table

A) A conducting rectangular loop falls from rest into a magnetic
field directed out of the page.
B) A thin metal strip slides down parallel rails elevated at an
angle q. A constant B field is directed vertically upward.
C) A conducting loop moves in a magnetic field produced by an
infinite current-carrying wire.

1. No situations
2. A
3. A and B
4. A and C
5. A, B, and C

### Commentary:

(4) There is usually a lot of confusion with Lenz’s Law. It is important
to determine what students are using to decide the direction of the
current. Some students despair of ever figuring it out and just guess.
Examining the microscopic motion of charges often helps.

# A2L Item 238

For which of the following is there an induced emf?

1. A conducting rod is pulled on conducting
rails that are placed in a uniform magnetic field directed into the
page.
2. A conducting loop moves through a uniform magnetic field
directed into the page.
3. A conducting loop rotates in a uniform
magnetic field directed into the page.
4. A conducting loop moves
in a magnetic field produced by an infinite current-carrying wire.
1. A only
2. A and B only
3. A and C only
4. A and D only
5. B and C only
6. A, B, and C
7. A, C, and D
8. All of them
9. None of the above