Electric Charge and Electric Force：練習問題

Two infinite, parallel non-conducting sheets carry surface charge densities \(+\sigma\) and \(-2\sigma\) respectively. The sheets are separated by a distance \(d\). If the sheet with \(+\sigma\) is located at \(z = 0\) and the sheet with \(-2\sigma\) is at \(z = d\), determine the electric field vector in the region \(0 < z < d\).

Consider an infinite non-conducting sheet with a uniform surface charge density \(\sigma\). A small circular hole of radius \(R\) is cut out of the sheet. Find the magnitude of the electric field at a point \(P\) located at a distance \(z\) directly above the center of the hole.

A non-conducting sphere of radius \(R\) carries a non-uniform volume charge density given by \(\rho(r) = \rho_0 \frac{r}{R}\) for \(r \le R\), and \(0\) otherwise. Using Gauss's Law, find the magnitude of the electric field at a distance \(r < R\) from the center.

A thin spherical shell of radius \(R\) has a total charge \(Q\) distributed uniformly over its surface. A point charge \(q\) is placed at a distance \(r = R/2\) from the center. What is the magnitude of the electrostatic force acting on the point charge \(q\)?

A point charge \(+q\) is located at the center of a cube of side length \(L\). What is the electric flux through one of the faces of the cube?

A non-conducting solid sphere of radius \(R\) has a non-uniform volume charge density given by \(\rho(r) = \rho_0 \frac{r}{R}\), where \(r\) is the distance from the center. Using Gauss's Law, find the expression for the electric field strength \(E\) at a point inside the sphere where \(r < R\).

A solid non-conducting sphere of radius \(R\) carries a non-uniform volume charge density given by \(\rho(r) = \rho_0 (1 - \frac{r}{R})\) for \(r \le R\), and \(\rho = 0\) for \(r > R\), where \(\rho_0\) is a positive constant.
(a) Use Gauss's Law to derive an expression for the electric field magnitude \(E\) as a function of distance \(r\) from the center for \(r \le R\).
(b) Determine the value of \(r\) at which the electric field intensity is maximum.