ap-Physics-2-2025-free-response-questions-and-answer/

Question 1

Very long Wire 1 carries current \(I\) in the \(+x\)-direction along the line \(y=0\). Very long Wire 2 carries current \(I\) in the \(+x\)-direction along the line \(y=+d\). Point \(P\) is located along Wire 1 at the origin, as shown in Figure 1. The diameters of the wires are small compared to the distance between the wires. Both wires are in the \(xy\)-plane.

(a)(i) Complete the following tasks in figures 2 and 3. Use either arrows or the symbols shown in the box above the figures for your response.
• Indicate the direction of the magnetic field from Wire 2 at Point \(P\) in Figure 2.
• Indicate the direction of the magnetic force that is exerted on Wire 1 by Wire 2 in Figure 3.

(a)(ii) Very long Wire 3 carrying current \(2I\) in the \(+x\)-direction is placed in the \(xy\)-plane along the line \(y=y_{3}\). The net magnetic force exerted on Wire 1 by the currents in wires 2 and 3 is zero. Derive an expression for \(y_{3}\) in terms of \(d\). Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

(b) Wire 3 is moved very far away from wires 1 and 2. A circular conducting loop in the \(xy\)-plane is initially held at rest below Wire 1. The loop is then moved at a constant speed in the \(-y\)-direction, as shown in Figure 4. Indicate whether there is a clockwise induced current in the loop, a counterclockwise induced current in the loop, or no induced current in the loop.

Most-appropriate topic codes (CED):

• TOPIC 4.2: Contact Forces and Fields (Magnetic Field of a Wire) — part (a-i)
• TOPIC 4.3: Magnetic Force (Force Between Current-Carrying Wires) — part (a-i, a-ii)
• TOPIC 4.4: Electromagnetic Induction (Lenz’s Law) — part (b)

▶️ Answer/Explanation

Detailed solution

(a)(i)
• Magnetic Field: Into the page (\(\otimes\)). Right Hand Rule: Thumb in \(+x\) (current), fingers curl into page below wire.
• Magnetic Force: Up (\(+y\)). Parallel currents attract; Wire 2 pulls Wire 1 upward.

(a)(ii)
Forces on Wire 1 must balance: \(\vec{F}_{2 \to 1} + \vec{F}_{3 \to 1} = 0\).
Wire 2 exerts an upward attractive force. Wire 3 must exert a downward attractive force.
For attraction, currents must be parallel (same direction), so Wire 3 must be located below Wire 1 (\(y_3 < 0\)).
Equating magnitudes:
\(\frac{\mu_0 I_1 I_2 L}{2\pi d} = \frac{\mu_0 I_1 I_3 L}{2\pi |y_3|}\)
\(\frac{I^2}{d} = \frac{I(2I)}{|y_3|} \Rightarrow \frac{1}{d} = \frac{2}{|y_3|}\)
\(|y_3| = 2d \Rightarrow\) \(y_3 = -2d\).

(b)
Clockwise.
The magnetic field from Wire 1 below the wire points into the page.
As the loop moves away (\(-y\)), the field strength and flux decrease.
Lenz’s Law: Induced field points into the page to oppose the decrease.
Right Hand Rule: Current must be clockwise.

Question 2

A sample of a monatomic ideal gas is sealed in a thermally conducting container by a movable piston of mass \(M\) and area \(A\). The container is in a large water bath that is held at a constant temperature \(T_{0}\). The piston is free to move with negligible friction. At the instant shown, the gas is in thermal equilibrium with the water bath, the piston is at rest, and the gas occupies volume \(V_{0}\). The pressure of the air above the piston is \(P_{atm}\).

(a) On the dot shown, representing the piston, draw and label the forces that are exerted on the piston. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot.

\(\bullet\)

(b) Derive an expression for the internal energy of the gas in terms of \(M\), \(A\), \(V_{0}\), \(P_{atm}\) and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

(c) A block, also of mass \(M\), is placed on the piston at time \(t=t_{0}\) and is slowly lowered. The piston comes to rest at \(t=t_{f}\) when the block is completely released.

On the axes provided, sketch the expected relationship between the pressure \(P\) and volume \(V\) of the gas for the thermodynamic process that the gas undergoes during time interval \(t_{0}\le t\le t_{f}\). Draw an arrow on your sketch to represent the direction of the thermodynamic process.

(d) With the block still on the piston, the temperature of the water bath is changed to a new constant temperature \(T_{new}\). The gas occupies the original volume \(V_{0}\) when the sample of gas and the water bath come to thermal equilibrium.

Indicate whether \(T_{new}\) is greater than, less than, or equal to \(T_{0}\).
\(\square \ T_{new}>T_{0}\)
\(\square \ T_{new}<T_{0}\)
\(\square \ T_{new}=T_{0}\)

Briefly justify your answer by referencing at least one feature of your answers to parts (a), (b), or (c).

Most-appropriate topic codes (CED):

• TOPIC 2.3: Thermodynamics and Forces — part (a)
• TOPIC 2.2: Pressure, Thermal Equilibrium, and the Ideal Gas Law — part (b)
• TOPIC 2.5: Thermodynamic Processes and PV Diagrams — part (c)
• TOPIC 2.2: Pressure, Thermal Equilibrium, and the Ideal Gas Law — part (d)

▶️ Answer/Explanation

Detailed solution

(a)
Forces acting on the piston:
• Upward: \(F_{gas}\) (Force from gas pressure)
• Downward: \(F_{g}\) or \(Mg\) (Weight of the piston) and \(F_{atm}\) (Atmospheric force)
Free body diagram of piston

(b)
Internal energy for a monatomic ideal gas: \(U = \frac{3}{2}nRT = \frac{3}{2}PV_0\)
From force equilibrium in part (a): \(P_{gas}A = P_{atm}A + Mg \Rightarrow P = P_{atm} + \frac{Mg}{A}\)
Substitute \(P\):
\(U = \frac{3}{2}V_{0}\left(P_{atm} + \frac{Mg}{A}\right)\)

(c)
The process is an isothermal compression (temperature held constant by water bath).
• Curve is concave up.
• Direction is up and to the left (Volume decreases, Pressure increases).
PV Diagram showing compression

(d)
\(T_{new} > T_{0}\)
Justification: The total mass on the piston has increased, so the gas pressure required for equilibrium is higher (\(P_{new} > P_{initial}\)). Since the volume is restored to \(V_0\) (constant) and \(PV = nRT\), a higher pressure \(P\) requires a higher temperature \(T\).

Question 3

In Experiment 1, a student is given a resistor of unknown resistance and an air-filled parallel-plate capacitor of unknown capacitance. The student is asked to predict the expected time constant \(\tau\) of a circuit if these two circuit elements were connected in series with a battery. The student has access to a battery of known emf, a switch, an ammeter, a ruler, and wires, as shown in Figure 1. The plates of the capacitor are square, and the separation between the plates is small compared to the dimensions of the plates. The capacitor is initially uncharged. Assume that the dielectric constant of air is \(1\).

[Figure 1: Circuit components]

(a) Describe a procedure for collecting data that would allow the student to determine the expected time constant \(\tau\). In your description, include the measurements to be made. Include any steps necessary to reduce experimental uncertainty.

(b) Describe how the collected data could be analyzed to determine \(\tau\). Include references to appropriate equations and to relationships between measured and known quantities.

(c) In Experiment 2, the student is asked to determine the capacitance \(C\) of a new parallel-plate capacitor. For each trial, the absolute value \(|\Delta V|\) of the potential difference across the capacitor is varied and the charge \(q\) stored on one plate of the fully charged capacitor is measured. Table 1 contains the data collected.

\(\|\Delta V\|\) (V)	\(3.0\)	\(5.0\)	\(7.2\)	\(8.0\)	\(10.0\)
\(q (\times 10^{-10} \text{C})\)	\(2.4\)	\(4.2\)	\(5.6\)	\(6.6\)	\(8.0\)

(i) Indicate two quantities, either measured quantities from Table 1 or additional calculated quantities, that could be graphed to produce a straight line that could be used to determine \(C\).
Vertical axis: ____________________
Horizontal axis: __________________

(ii) On the grid provided, create a graph of the quantities indicated in part (c)(i). Clearly label the axes, including units as appropriate. Plot the points you recorded in Table 2.

(iii) Draw a best-fit line for the data plotted in part (c)(ii).

(iv) Using the best-fit line, determine the capacitance \(C\) of the capacitor. Show your work.

Most-appropriate topic codes (CED):

• TOPIC 3.10: Capacitance and Geometry of Parallel Plates — part (a, b)
• TOPIC 3.8: Ohm’s Law and RC Circuits — part (a, b)
• TOPIC 3.12: Capacitance (Q=CV relationship) — part (c)

▶️ Answer/Explanation

Detailed solution

(a)
Procedure:
1. Use the ruler to measure the side length \(s\) of the capacitor plates and the plate separation \(d\). Repeat measurements to find an average.
2. Construct a circuit with the battery, switch, resistor, and ammeter (capacitor not needed for \(R\)).
3. Close the switch and measure the current \(I\).

(b)
Analysis:
1. Calculate capacitance \(C = \frac{\epsilon_0 s^2}{d}\).
2. Calculate resistance \(R = \frac{\mathcal{E}}{I}\) using the battery emf \(\mathcal{E}\).
3. Calculate time constant \(\tau = RC\).

(c)(i)
Vertical axis: Charge \(q\) (or \(|\Delta V|\))
Horizontal axis: Potential Difference \(|\Delta V|\) (or \(q\))

(c)(ii) & (iii)
The graph of \(q\) versus \(|\Delta V|\) with the best-fit line:
0246810|ΔV| (V)02468q (x10^-10 C)

(c)(iv)
\(C\) is the slope of the line \(q = C|\Delta V|\).
Using points on the line \((0,0)\) and \((10.0, 8.0)\):
Slope \(= \frac{8.0 – 0}{10.0 – 0} \times 10^{-10} = 0.8 \times 10^{-10}\)
\(C = 8.0 \times 10^{-11} \text{F}\).

Question 4

Two narrow slits are a distance \(d\) apart. A screen is a distance \(L\) from the midpoint of the slits, where \(L\gg d\). When a laser emits monochromatic light toward the slits, a pattern of narrow bright and dark bands is observed on the screen. The centers of bright bands A and B are indicated. Three additional bright bands, including the central bright band, are observed on the screen between bands A and B, as shown.

A student claims that the distance between the center of Band A and the center of the central bright band is smaller when using a laser that emits violet light than when using a laser that emits red light.

(a) Indicate whether the student’s claim is correct or incorrect. Without manipulating equations, justify your answer by referencing the difference in path length traveled by the light from each slit to the center of Band A.

(b) Derive an expression for the distance between the centers of bands A and B when light of frequency \(f\) is emitted toward the slits. Express your answer in terms of \(d\), \(L\), \(f\), and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

(c) Indicate whether the expression you derived in part (b) is or is not consistent with your answer from part (a). Briefly justify your answer.

Most-appropriate topic codes (CED):

• TOPIC 6.6: Interference and Diffraction — part (a), (b), (c)

▶️ Answer/Explanation

Detailed solution

(a)
Correct.
Violet light has a shorter wavelength \(\lambda\) than red light. Constructive interference occurs when the path length difference \(\Delta \ell = m\lambda\). A shorter \(\lambda\) requires a shorter path difference \(\Delta \ell\) for the same order \(m\). A shorter \(\Delta \ell\) corresponds to a smaller distance from the central bright band.

(b)
\(d \sin \theta = m \lambda\)
Using small angle approximation \(\sin \theta \approx \frac{y}{L}\):
\(d \frac{y}{L} = m \lambda \Rightarrow y = \frac{m \lambda L}{d}\)
Band A is the second bright fringe (\(m=2\)) and Band B is the symmetric fringe (\(m=-2\)).
Distance \(= y_A – y_B = \frac{2 \lambda L}{d} – \frac{-2 \lambda L}{d} = \frac{4 \lambda L}{d}\)
Substitute \(\lambda = \frac{c}{f}\):
Distance \(= \frac{4cL}{fd}\)

(c)
Consistent.
The derived expression \(\frac{4cL}{fd}\) shows the distance is inversely proportional to frequency \(f\). Violet light has a higher frequency than red light, resulting in a smaller distance.

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