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.

Describe a procedure the students could use to create a reproducible fixed amount of charge on the capacitor plates using the two identical conducting spheres and then collect data needed to determine the capacitor’s capacitance. Include in your procedure how charge is distributed in a conductor at electrostatic equilibrium, how charge moves when conductors contact, and at least one step that reduces experimental uncertainty.

Describe how the students could use their measurements to graphically determine the capacitance $C$ of the parallel-plate capacitor and how the graph would be analyzed. In your description, specify what would be plotted on each axis and how a feature of the graph (slope or intercept) is related to $C$ through an equation.

Using the best-fit line from part C(iii) (or an equivalent relationship consistent with your graph), calculate an experimental value for the capacitance $C$ in air. Then calculate the expected potential difference $|\Delta V|$ when the dielectric is inserted and the charge is unchanged. For one run, the students set $d = 0.0040\ \text{m}$ and measure $|\Delta V| = 30.6\ \text{V}$. The plate area is $A = (0.120\ \text{m})^2$. Assume the capacitor is in air so $\kappa = 1.00$ and $\varepsilon_0 = 8.85× 10^{-12}\ \text{F/m}$. In a second run with the same plate separation, the students insert a dielectric sheet that completely fills the space between the plates. The dielectric has $\kappa = 3.2$, and the capacitor is isolated so the charge on the plates remains the same.

In Figure 2, draw bars to represent the relative magnitudes of (i) the charge |Q| on each sphere and (ii) the electric potential |V| of each sphere for the initial and final states. Use the given bar as a reference. If a quantity is zero, write a "0" in that column. After the wire is connected and electrostatic equilibrium is reached, the wire is removed. The partially completed bar chart in Figure 2 shows a bar that represents the magnitude of the electric potential |V| of sphere 2 in the final state.

Derive an expression for the final charge on sphere 2, $Q_{2f}$, after the wire is removed, in terms of $Q_0$, $R_1$, $R_2$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Sphere 2 is then touched to (and thereby charges) the isolated top plate of the parallel-plate capacitor, and the bottom plate remains grounded, so the capacitor has charge magnitude $|Q| = |Q_{2f}|$ on its plates. A dielectric slab of thickness $d = 1.0\ \text{mm}$ and dielectric constant $\kappa = 3.0$ is slowly inserted between the plates from one side so that it fills a fraction $x/d$ of the plate area, where $x$ is the insertion depth measured in meters and $0 ≤ x ≤ d$. On the axes shown in Figure 3, sketch a graph of the potential difference $V$ across the capacitor as a function of $x$.

Indicate whether the graph you sketched in part C is or is not consistent with your bars in part A. Briefly justify your answer by referencing the functional dependence between $V$ and the capacitance $C$ for a capacitor with fixed charge, and how the effective capacitance changes as the dielectric is inserted. Use $\varepsilon_0 = 8.85× 10^{-12}\ \text{F/m}$. The capacitor has $A = 0.020\ \text{m}^2$ and $d = 1.0\ \text{mm}$. The dielectric constant is $\kappa = 3.0$. The initial charge on sphere 1 is $Q_0 = 12\ \text{nC}$, and the sphere radii are $R_1 = 2.0\ \text{cm}$ and $R_2 = 6.0\ \text{cm}$. Assume the capacitor remains isolated after being charged, so the charge magnitude on the plates stays equal to $|Q_{2f}|$ while the dielectric is inserted.