AP Physics C: Electricity & Magnetism 10 Conductors and Capacitors Questions

A nonconducting sphere of uniform volume charge density is surrounded by a thin concentric conducting spherical shell, as shown in the cutout view. The sphere has a charge of -Q and the shell has a charge of +3 Q. The radii of the inner sphere and spherical shell are R and 4 R, respectively, as shown in the cross-section view.

Determine the charge on the outer surface of the shell.

Conducting spheres 1 and 2 on insulating stands have radii 2 R and R, respectively. The spheres are very far apart. Sphere 1 has an initial net charge +Q and Sphere 2 is initially neutral. The spheres are then connected by a conducting wire, as shown. The wire is then removed. Which of the following represents the value of the net charge on Sphere 1 after the wire is removed?

The plates of a certain variable capacitor have an adjustable area. An experiment is performed to study the potential difference across the capacitor as it discharges through a resistor. A circuit is to be constructed with the following available equipment: a single ideal battery of potential difference $\Delta V_{0}$, a single voltmeter, a single resistor of resistance R, a single uncharged variable capacitor set to capacitance C, and one or more switches as needed.

The capacitor is adjusted so that the surface area of the plates is increased, and the experiment is repeated. Would the slope of the best-fit line in the second experiment be more steep, less steep, or unchanged compared to the slope of the best-fit line in part (c)? More steep Less steep Unchanged

Briefly justify your answer.

An isolated, air-filled, charged capacitor consists of two conducting, coaxial, cylindrical shells that each have length L. The inner shell has radius $R_{1}$ and the outer shell has radius $R_{2}$, as shown in Figure 1, where $R_{1}<R_{2} \ll L$. The surface charge densities (amounts of charge per unit area) of the inner and outer shells are $+\sigma_{1}$ and $-\sigma_{2}$, respectively. The absolute values of the total charges on the shells are equal.

Note: Figures not drawn to scale.

A material of dielectric constant $\kappa$ is inserted into the isolated, charged capacitor such that the material fills the region $R_{1}<r<R_{2}$, as shown in Figure 3.

Derive an expression for the capacitance C of the capacitor with the material inserted in terms of $L, R_{1}, R_{2}, \kappa$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.