AP Physics C: Electricity and Magnetism Unit 10 Review: Conductors and Capacitors

Sphere A and Sphere B are brought into contact with each other and then separated so they are far apart again.

Derive an expression for the magnitude of the electric field $E$ at a point on the surface of Sphere A after the spheres have been separated, in terms of $Q_A$, the sphere radius $R = 2.0\ \text{cm}$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Indicate whether $Q_\text{new}$ is greater than, less than, or equal to $Q_0$ by writing one of the following. The parallel-plate capacitor (plate area $A = 0.020\ \text{m}^2$, separation $d = 1.0\ \text{mm}$) is connected to a $12\ \text{V}$ ideal battery. The capacitor is initially filled with air. The dielectric slab with dielectric constant $\kappa = 4.0$ is then inserted completely between the plates while the capacitor remains connected to the battery, as shown in Figure 2. Let $Q_\text{new}$ be the new magnitude of the charge on each plate after insertion and let $Q_0$ be the original magnitude of the charge on each plate before insertion.

In Figure 2, draw bars to represent $|E|$ at $r=1.0\ \text{mm}$, $3.0\ \text{mm}$, and $8.0\ \text{mm}$ relative to the $|E|$ shown at $r=4.0\ \text{mm}$. If $E=0$, write a "0" in that column. Let $E(r)$ be the magnitude of the electric field at a distance r from the axis. The partially completed bar chart in Figure 2 shows a bar that represents $|E|$ at $r=4.0\ \text{mm}$.

Derive an expression for the electric field magnitude $E(r)$ in the region $a<r<b$ in terms of $\lambda$, $\varepsilon$, $r$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Consider a coaxial cylindrical Gaussian surface of length $L = 0.20\ \text{m}$ and radius $r$, centered on the axis as in Figure 1. On the axes shown in Figure 3, sketch a graph of the electric flux $\Phi_E(r)$ through this closed Gaussian surface as a function of r for $0<r<10\ \text{mm}$.

Indicate whether the graph you drew in part C is or is not consistent with the potential difference $\Delta V = V(a)-V(b)$ you would obtain by evaluating $\Delta V = -\int_b^a E(r)\,dr$. Briefly justify your answer by referencing the functional dependence between $\Phi_E(r)$ and the enclosed charge and by stating whether $\Delta V$ should be positive, negative, or zero. A student uses the result of part B to compute the potential difference between the conductors. Use $a = 2.0\ \text{mm}$, $b = 6.0\ \text{mm}$, $\lambda = 4.0× 10^{-9}\ \text{C/m}$, $\varepsilon = 2.5\varepsilon_0$, and $\varepsilon_0 = 8.85× 10^{-12}\ \text{F/m}$. The potential of the outer conductor is defined to be $V(b)=0\ \text{V}$.

Describe a procedure for collecting data that would allow the students to use a graph to determine the plate area $A$. Your procedure should include how the students would measure all relevant quantities, how the capacitor should be charged and isolated for each trial, and how the students would ensure that the inserted conducting slab remains electrically isolated. Include at least one step that would reduce experimental uncertainty. In your description, make clear how charge is distributed in the conductor at electrostatic equilibrium and what would happen if the slab were to touch one of the plates.

Describe how the collected data could be graphed and how that graph would be analyzed to determine the plate area $A$. Your description should specify which quantity is placed on each axis and how a feature of the graph is related mathematically to $A$ using the relationship for an isolated conducting slab between the plates, $C=\frac{\epsilon_0 A}{D-t}$, where $\epsilon_0=8.85\times10^{-12}\ \mathrm{F/m}$. Figure 2 shows the charge distribution on the capacitor plates and slab surfaces at electrostatic equilibrium after the power supply has been disconnected.

Using the best-fit line that you drew in part C(iii), calculate an experimental value for the plate area $A$. Use the best-fit line from part C(iii) and the relationship $C=\frac{Q}{V}=\frac{\epsilon_0 A}{D-t}$. The plate separation is $D=3.00\times10^{-3}\ \mathrm{m}$, the charging potential difference is $V=100\ \mathrm{V}$, and $\epsilon_0=8.85\times10^{-12}\ \mathrm{F/m}$.