Practice Test 11 - AP Physics C Electricity and Magnetism

20 questions. Answer them in any order, then submit to see your score and every explanation.

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1. A uniform electric field is given by $\vec{E} = E_0 \hat{k}$. What is the electric flux through the curved side surface of a cylinder of radius $R$ and height $H$, whose axis is aligned with the z-axis?
2. A square surface with side length $L$ is in a uniform electric field $\vec{E}$. The electric field vector makes an angle of 60° with the normal to the surface. What is the magnitude of the electric flux through the surface?
3. A polymer sphere of radius $R=0.050\ \text{m}$ is manufactured with uniform volume charge density $\rho=+2.0\ \mu\text{C/m}^3$ to create a known calibration field. A probe is placed at $r=0.030\ \text{m}$ from the center (inside the sphere). Consider the setup described above. The field is needed to verify Gauss's law behavior in the bulk material. Assume spherical symmetry and express the field in N/C with direction $\pm\hat{r}$.
4. A student correctly applies Kirchhoff's junction rule, $\sum I_k = 0$, to a node in a complex DC circuit. Which of the following statements provides the most fundamental justification for the validity of this rule in this context?
5. A dielectric material is inserted between the plates of an isolated parallel-plate capacitor that is initially charged. Which of the following best describes what happens to the electric field between the plates?
6. A capacitor of capacitance $C$ is charged to a potential difference $E$ by a battery through a resistor $R$. What is the total energy dissipated as heat by the resistor during the entire charging process?
7. The electric field in a certain region of space is given by $\vec{E} = C r^2 \hat{r}$, where $C$ is a positive constant and $\hat{r}$ is the radial unit vector. What is the total charge enclosed within a spherical surface of radius $R$ centered at the origin?
8. A constant potential difference of 12 V is maintained across a 4.0 mH inductor. What must be the rate of change of current through the inductor?
9. A battery with emf $\varepsilon=15\,\text{V}$ and internal resistance $r=0.50\,\Omega$ is connected in series with an external resistor $R=4.5\,\Omega$. Using the loop rule, what is the potential difference across the battery terminals during operation?
10. Electric charge is a conserved property of matter measured in coulombs (C); it comes in positive and negative types, and the total charge of an isolated system remains constant. Electric forces act at a distance: like charges repel, unlike charges attract, and the force magnitude depends on charge and separation. In space, two point charges $q_1=+4.0\times10^{-6}\ \text{C}$ and $q_2=+1.0\times10^{-6}\ \text{C}$ are fixed $0.30\ \text{m}$ apart. Coulomb's Law gives the magnitude: $F=k\frac{|q_1q_2|}{r^2}$. A numerical check: $F=(8.99\times10^{9})\frac{(4.0\times10^{-6})(1.0\times10^{-6})}{(0.30)^2}$. If a third charge is later introduced, the principle of superposition states the net force on any charge equals the vector sum of forces from each other charge calculated separately. How does the force between charges change if the distance is doubled?
11. A uniform magnetic field $\vec{B} = B_0 \hat{k}$ exists in space. What is the magnitude of the magnetic flux through the curved surface of a hemisphere of radius $R$ whose circular base lies in the xy-plane and is centered at the origin?
12. An electrostatic sensor is tested near a uniformly charged thin ring of radius $R=0.30\ \text{m}$ carrying total charge $Q=+2.0\ \text{nC}$. The sensor is placed far on the axis at $z=3.0\ \text{m}$ to check the far-field approximation. Consider the setup described above. The field is needed to confirm the ring behaves like a point charge at large distances. Use the ring's on-axis expression and indicate direction along $\pm\hat{z}$ in N/C.
13. A DC motor with winding $L=40\,\text{mH}$ and $R=2\,\Omega$ is switched off; current was $3\,\text{A}$; Refer to the scenario above. Explain the significance of Lenz's Law in the scenario provided.
14. In a series RC circuit with time constant $τ$, an initially uncharged capacitor is being charged by a battery with emf $E$. At what time $t$ will the potential difference across the capacitor be equal to half of the battery's emf?
15. A long solenoid of radius $R$ has $n$ turns per unit length and carries a current $I$. The magnetic field inside is uniform and given by $B=\mu_0 n I$. A smaller, coaxial circular loop of radius $r < R$ is placed inside the solenoid. What is the magnetic flux through the smaller loop?
16. The electric potential energy $U$ of a uniformly charged conducting sphere of radius $R$ and total charge $Q$ is $U = \frac{Q^2}{8\pi\epsilon_0 R}$. This can be interpreted as the energy stored in a capacitor. What is the effective self-capacitance of the sphere?
17. Electric charge is a conserved property of matter measured in coulombs (C); it can be positive or negative, and interactions between charges produce electric forces. Electric force acts along the line joining charges and follows Coulomb's Law $F=k\frac{|q_1q_2|}{r^2}$. In space, two point charges $q_1=+6.0\times10^{-6}\ \text{C}$ and $q_2=-1.0\times10^{-6}\ \text{C}$ are separated by $r=0.30\ \text{m}$. Numerical setup: $F=(8.99\times10^{9})\frac{|(6.0\times10^{-6})(-1.0\times10^{-6})|}{(0.30)^2}$. If additional charges are introduced, the net force is found by superposition, adding the individual force vectors. Calculate the force between two charges of $+6.0\ \mu\text{C}$ and $-1.0\ \mu\text{C}$, separated by $0.30\ \text{m}$.
18. Electric charge is a conserved property (C) that can be positive or negative; like charges repel and unlike charges attract. Electric force between point charges acts along the line joining them and obeys Coulomb's Law. Three point charges lie on a line: $q_1=+2.0\times10^{-6}\ \text{C}$ at $x=0$, $q_2=-1.0\times10^{-6}\ \text{C}$ at $x=0.30\ \text{m}$, and $q_3=+2.0\times10^{-6}\ \text{C}$ at $x=0.60\ \text{m}$. For $q_2$, the forces from $q_1$ and $q_3$ have magnitudes $F_{21}=k\frac{|q_2q_1|}{(0.30)^2},\quad F_{23}=k\frac{|q_2q_3|}{(0.30)^2}$ and superposition requires adding these vectors to find the net force. Determine the net electric force on $q_2$ placed in an electric field.
19. The conventional current in a horizontal wire is directed from east to west. What is the direction of the drift velocity of the free electrons in the wire?
20. A proton ($q=+e=+1.60\times10^{-19}\ \text{C}$) moves from point $X$ at $V_X=+250\ \text{V}$ to point $Y$ at $V_Y=-50\ \text{V}$. Use $\Delta U=q\Delta V$ and $W_{\text{field}}=-\Delta U$ (and $k=8.99\times10^{9}$). Based on the data, determine the work done moving the proton from $X$ to $Y$ by the electric field.

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