AP Physics C E and M 8.1: Electric Charge and Force

Describe electric charge and the electric force between charged objects or systems, relating force magnitude and direction to charge interactions.

[Maximum number: 11]

Students perform an experiment to study the force between two charged objects using the apparatus shown above, which contains two identical conducting spheres. The upper sphere is attached to an insulating string, which can be used to move the sphere downward. The lower sphere sits on an insulating rod, which is on an electronic balance. The electronic balance is zeroed before the lower sphere and insulating rod are in place.
For the first trial, a charge of Q is placed on each sphere and then the upper sphere is slowly moved downward. The students measure the distance d between the centers of the spheres and the magnitude F of the force that appears on the electronic balance. The recorded data are shown on the graph of F as a function of $\frac{1}{d^{2}}$ shown below.

Figure for Question 8.1 Electric Charge and Electric Force question 1 — AP Physics C: Electricity & Magnetism

(a)

Draw a line that represents the best fit to the points shown.

Use the graph to calculate the charge Q.

iii. On the graph on the previous page, draw a circle around the data point that was taken when the distance between the centers of the spheres was the least.
iv. Determine the distance between the centers of the spheres for the data point indicated above.
v. What physical quantity does the vertical intercept represent?

Justify your answer.

Figure for Question (a) — AP Physics C: Electricity & Magnetism

The experiment is extended by collecting additional data points, which appear on the right side of the graph shown above. The new data points do not follow the linear pattern seen with the first points. The group of students tries to explain this discrepancy.

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i. For drawing an appropriate best-fit line 1 point
Example response for part (a)(i)

Figure for Answer to Question (a) — AP Physics C: Electricity & Magnetism

For calculating the slope using two points from the best-fit line 1 point

\text { slope }=m=\frac{\Delta y}{\Delta x}=\frac{(12,000-10,500) \mu \mathrm{N}}{(72-40) \mathrm{m}^{-2}}=4.7 \times 10^{-5} \mathrm{~N} \cdot \mathrm{~m}^{2}

For correctly relating the slope to the charge 1 point

F=\frac{k Q^{2}}{r^{2}} \therefore \text { slope }=k Q^{2}

For correctly calculating the charge consistent with the slope from the best-fit line 1 point

Q=\sqrt{\frac{\text { slope }}{k}}=\sqrt{\frac{\left(4.7 \times 10^{-5} \mathrm{~N} \cdot \mathrm{~m}^{2}\right)}{\left(9 \times 10^{9} \mathrm{~N} \cdot \mathrm{~m}^{2} / \mathrm{C}^{2}\right)}}=7.2 \times 10^{-8} \mathrm{C}

iii. For drawing a circle around the appropriate data point 1 point

For a correct answer 1 point
From the graph, $\frac{1}{d^{2}}=82.5 \mathrm{~m}^{-2}$ $d=0.11 \mathrm{~m}$

For indicating that the y-intercept should be the weight of the conducting sphere 1 point and insulating rod
For a correct justification 1 point
Example responses for part (a)(v).
When the spheres are infinitely far apart, there is no electric force between them.
The only force recorded by the balance must be the weight of the lower sphere and insulating rod.
OR
The scale is zeroed before the insulating rod, and lower sphere are placed on it, so the nonzero y-intercept is their weight.

Total for part (a)

8 points

(b)

A third student suggests that the experiment be modified so that the top sphere is given a negative charge that is equal in magnitude to the positive charge given to the bottom sphere.

On the circles representing the spheres below, use a single "+" sign on the bottom sphere to represent the location of highest concentration of the excess positive charges. Use a single "-" sign on the top sphere to represent the location of the highest concentration of the excess negative charges.

For a separation distance equal to that of the data point indicated in part (a)(iii), would the magnitude of the force reading with spheres of opposite charges be greater than, less than, or equal to the magnitude of the force reading with spheres of the same charges?

Greater than Less than Equal to
Justify your answer.

Figure for Question (b) — AP Physics C: Electricity & Magnetism

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[Maximum number: 1]

32. A proton p (charge +e, mass $m_{p}$ ) collides head-on with a deuteron d (charge +e, mass $2 m_{p}$ ). During the collision the particles interact through an electrostatic Coulomb force. Which of the following statements is true about the accelerations $\vec{a}_{p}$ of the proton and $\vec{a}_{d}$ of the deuteron during the collision?

A

$\vec{a}_{p}=-2 \vec{a}_{d}$

B

$\vec{a}_{p}=-\vec{a}_{d}$

C

$\vec{a}_{p}=-\vec{a}_{d} / 2$

D

$\vec{a}_{p}=\vec{a}_{d}$

E

$\vec{a}_{p}=2 \vec{a}_{d}$

Figure for Question 8.1 Electric Charge and Electric Force question 2 — AP Physics C: Electricity & Magnetism

Top View

[Maximum number: 2]

Students design an experiment to determine the unknown dielectric constant $\kappa$ of a plastic material.

A capacitor is created using two square aluminum plates of side length $s=30 \mathrm{~cm}$ that are separated by

a distance $d=1.0 \mathrm{~mm}$ . This capacitor is placed in a circuit with an ideal 6.0-volt battery, a resistor of resistance

$R=500 \Omega$ , voltmeter V, and an open switch S, as shown above. A1.0 mm thick piece of plastic is inserted

between the aluminum plates. The distance x that the plastic is inserted between the plates can be varied, and the

voltmeter is used to measure the potential difference $V_{C}$ across the capacitor. The switch is closed, and readings

from the voltmeter are recorded as a function of time t. The data are plotted to create the graph shown below.

Figure for Question 8.1 Electric Charge and Electric Force question 3 — AP Physics C: Electricity & Magnetism

The time $t_{1 / 2}$ shown above is the time for the capacitor to charge to half the potential difference of the battery.

The students now want to verify the value for the permittivity constant, $\varepsilon_{0}$ . Using the graph from part (b),

calculate an experimental value for $\varepsilon_{0}$ .

CNV-7.A.a, SP 4.D, 6.C

2 points

The students now want to verify the value for the permittivity constant, $\varepsilon_{0}$ . Using the graph from

part (b), calculate an experimental value for $\varepsilon_{0}$ .

\begin{tabular}[t]{|l|l|}
\hline For using an acceptable value for the y-intercept consistent with the best-fit line

1 point

For correctly relating the y-intercept to the permittivity constant

1 point

$C=\frac{\varepsilon_{0} s^{2}}{d}+\frac{\varepsilon_{0} s(\kappa-1)}{d} x \therefore$ y-intercept $=\frac{\varepsilon_{0} s^{2}}{d}$ $\varepsilon_{0}=\frac{(d)(\mathrm{y} \text {-intercept })}{s^{2}}=\frac{\left(1 \times 10^{-3} \mathrm{~m}\right)\left(8.4 \times 10^{-10} \mathrm{~F}\right)}{(.3 \mathrm{~m})^{2}}=9.29 \times 10^{-12} \mathrm{C}^{2} /\left(\mathrm{N} \cdot \mathrm{m}^{2}\right)$

Units point

CNV-7.A.a, SP 7.A1 point

\begin{tabular}[t]{|l|l|}
\hline For correct units in parts (d) and (e)

1 point

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Covered syllabus points

8.1 Electric Charge and Electric Force question 1

Question (a)

Question (b)

8.1 Electric Charge and Electric Force question 2

8.1 Electric Charge and Electric Force question 3