Electrostatics with Conductors：練習問題

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.

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?

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.