AP (Advanced Placement) · AP Physics C: Electricity and Magnetism
5 multiple-choice questions marked as you go, and 3 written questions with worked solutions. All on Electrostatics with Conductors.
A parallel-plate capacitor with plate area \(A\) and separation \(d\) is filled with two slabs of dielectric material. Slab 1 has a dielectric constant \(\kappa_1\) and a thickness of \(d/3\). Slab 2 has a dielectric constant \(\kappa_2\) and a thickness of \(2d/3\). Both slabs have the same area \(A\) as the plates. What is the equivalent capacitance of this arrangement?
Two identical capacitors are connected in parallel and charged to a potential difference \(V\). They are then disconnected from the source. A dielectric slab of constant \(\kappa\) is inserted into one of the capacitors, completely filling the space between its plates. What is the new potential difference across the capacitors?
A parallel-plate capacitor with plate area \(A\) and separation \(d\) is filled with two different dielectric materials. The first dielectric has a dielectric constant \(\kappa_1\) and thickness \(d/2\), and the second has a dielectric constant \(\kappa_2\) and thickness \(d/2\). If the plates are held at a potential difference \(V\), what is the total capacitance of the system?
A capacitor is composed of two long coaxial cylindrical conductors of radii \(a\) and \(b\) (\(b > a\)) and length \(L\). The space between the cylinders is filled with a material whose dielectric constant varies radially as \(\kappa(r) = \kappa_0 \frac{r}{a}\). What is the capacitance of this device?
A spherical conductor of radius \(R_1\) carries a charge \(Q\). It is surrounded by a concentric neutral conducting spherical shell with inner radius \(R_2\) and outer radius \(R_3\). What is the potential at a distance \(r\) from the center, where \(R_1 < r < R_2\)?
An isolated capacitor system consists of two concentric spherical conducting shells of radii \(a\) and \(b\) (where \(b > a\)). If the space between the shells is filled with a dielectric material of constant \(K\), derive the formula for the capacitance \(C\) and determine how the stored energy changes if the dielectric is removed while the charge \(Q\) remains constant.
Write your answer out first, then check it against the worked solution.
A hollow spherical conductor has an inner radius \(a\) and an outer radius \(b\). A point charge \(+q\) is placed at the center of the hollow space.
(a) Determine the surface charge density \(σ\) on the inner surface (\(r = a\)) and the outer surface (\(r = b\)).
(b) Calculate the electric field magnitude for \(r < a\), \(a < r < b\), and \(r > b\).
(c) If the outer surface is now grounded, how do the results for parts (a) and (b) change?
Write your answer out first, then check it against the worked solution.
A capacitor is formed by two coaxial cylindrical conductors of radii \(a\) and \(b\) (where \(a < b\)) and length \(L\), where \(L ≫ b\). The space between the cylinders is filled with air.
(a) Determine the capacitance \(C\) of this cylindrical capacitor.
(b) If the inner cylinder is given a charge \(+Q\) and the outer cylinder is grounded, find the total energy stored in the electric field between the cylinders.
Write your answer out first, then check it against the worked solution.
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