# A2L Item 214

Goal: Hone the concept of flux

Source: 283-400, Flux in and out of a balloon.

We construct a closed Gaussian surface in the shape of a sphericalWe construct a closed Gaussian surface in the shape of a spherical
balloon. Assume that a small glass bead with total charge Q is in the
vicinity of the balloon. Consider the following statements:

1. If the bead is inside the balloon, the electric flux over the
balloon’s surface can never be 0.

2. If the bead is outside the balloon,
the electric flux over the balloon surface must be 0.

Which of these statements is valid?

1. Only A is valid.
2. Only B is valid.
3. Both A and B are valid.
4. Neither one is valid.

### Commentary:

(3) Students may accept statement A but still think that the value of
the flux depends upon location of the bead in the sphere. Transition
from just inside to just outside poses particular difficulty to some
students. This usually derives from lack of experience with vectors and
dot products. Having the student draw field lines does help, but only
after they comprehend that the formal definition of flux is equivalent
to counting the net number of lines of E crossing the surface.

# A2L Item 213

Goal: Reasoning regarding electric fields due to distributed charges

Source: 283-395 Electric field from a rod, on its axis.

A rod of length L and charge +q
(uniformly distributed) is positioned along the x-axis, as shown to the
right. What is E at point P, a distance a from the origin?

 1 2 3 4 5

### Commentary:

(3) It is worthwhile having students examine their choice for the
limiting case a->0. Students are inclined to immediately start a formal
calculation rather than think about the problem long enough to figure
out what they really need to know. In this case all but two of the
answers can be ruled out because they do not limit appropriately as the
point P moves toward the origin. If a>>L the field should drop off as
from a point charge. The only answer meeting both these requirements is
3.

# A2L Item 212

Goal: Reason with electric fields

Source: 283-370 E due to circular rods

All of the curved charged rods shown in the image below have the same
radius and linear charge density (though some are positively charged and
others are negatively charged). For which configuration would the
magnitude of E at the origin be greatest?

### Commentary:

(6) This problem constitutes a good exercise for students learning the
vector nature of the electric field. There are many good followup
questions, such as; Which configurations have zero field at the origin?,
Order the configurations by increasing magnitude of electric field at
the origin. Stress the value of symmetry for reasoning to the answer. A
negative distribution in a quadrant is equivalent to a positive
distribution in the opposite quadrant, which means that distributions #5
and #7 are equivalent (for purposes of finding the E field at the
origin).

# A2L Item 210

Goal: Relate representations

Source: 283-342 Graph Ex(x)

We have a charge configuration
(shown at the right). Which graph is the plot of Ex(x), the
x component of the electric field, as you move along the
x-axis?

### Commentary:

(6) Students should recognize that the field goes singular at the
charges. The only graphs doing that, #1 and #4, are eliminated because
the x-component of the field must be negative everywhere between the two
charges. Have students sketch the graph.

# A2L Item 209

Goal: Translate among representations

Source: 283-341 graph Ex(y)

We have a charge configuration
(shown at the right). Which graph below resembles the plot of the x
component
of the electric field, Ex(y), as you move
along the y-axis?

### Commentary:

(6) The correct graph looks like the negative of graph #5. Some students
pick #5 thinking that the magnitude of the field is desired.

# A2L Item 208

Goal: Hone the concept of electric field

Source: 283-340 Where is E zero near a dipole? 9/21

Where,
other than at infinity, is the electric field 0 in the vicinity of the
dipole shown?

1. Along the y-axis.
2. At the origin.
3. At two points, one to the right of (a, 0), the other to the left of (-a,
0).
4. At two points on the y-axis, one below the origin, one above the origin.
5. None of the above.

### Commentary:

(5) Students have a lot of difficulty distinguishing electric field from
potential. Students already exposed to the concept of potential
frequently respond that the field is zero along the y-axis. If there are
many confused students, before identifying the correct response, it
helps to have the students draw the field lines.

# A2L Item 205

Goal: Hone the concept of a conductor

Source: 283-230 E in the cavity of a neutral conductor

A positive charge +Q is placed outside a neutral conductor. Inside the
conductor is a cavity containing no charge. What is the electric field
at the “center” of the cavity, a distance D from the charge +Q?

1. kQ/D2, direction away from the charge Q
2. kQ/D2, direction toward the charge Q
3. kQ/D2, direction away from the center of the conductor
4. kQ/D2, direction toward the center of the conductor
5. E = 0
6. Cannot determine the electric field

### Commentary:

(5) Induced charges and resulting fields pose problems for most
students. Having students compare this case to one where the charge is
inside the cavity helps. They tend to get hung up on the problem of
envisioning the exact distribution of induced charge that will cancel
the field everywhere inside the conductor.

# A2L Item 204

Goal: Hone the concept of a conductor

Source: 283-225, Charge induced in a neutral conductor

A positive charge +Q is placed outside a neutral conductor. Inside the
conductor is a cavity containing no charge. What is the net charge on
the surface bounding the cavity?

1. A positive charge +Q
2. A positive charge +q < +Q
3. Zero charge
4. A negative charge |-q| < |+Q|
5. A negative charge -Q
6. Cannot determine the charge.

### Commentary:

(3) Students have difficulty envisioning induced charge distributions,
especially when the conductor has an irregular shape.

# A2L Item 203

Goal: Reason regarding electric fields

Source: 283 –

Charge, Q, is at the origin. Points A-E are positions where other
charges may be present. Etotal at point F is non-zero and
points in the +i direction. Which of the following situations
could account for this?

1. Another charge is present at position
A.
2. Two other charges are present at C & D.
3. Two other
charges are present at D & B.
4. Two other charges are present at
E & A.
1. A only
2. B only
3. C only
4. D only
5. A and B
6. A, C and D
7. A, B, and D
8. A, B, C, and D

### Commentary:

(2) Only a charge at C can counter the y component of the field of Q at
F. A charge at B or D can then create a field at F that points in the i
direction.

# A2L Item 202

Goal: Reason regarding electric fields

Source: 283-11, E at origin due to charged rods

For which of the configuration(s) below does the total electric field
vector at the origin have non-zero components in both the x and y
directions?

1. 2 only
2. 1 and 3 only
3. 5 only
4. 4 only
5. 1 and 5 only
6. None of the above