6.3 Applying gauss’s law  (Page 10/13)

A very long, thin wire has a uniform linear charge density of $50\mu \text{C/m}\text{.}$ What is the electric field at a distance 2.0 cm from the wire?

A charge of $\mathrm{-30}\mu \text{C}$ is distributed uniformly throughout a spherical volume of radius 10.0 cm. Determine the electric field due to this charge at a distance of (a) 2.0 cm, (b) 5.0 cm, and (c) 20.0 cm from the center of the sphere.

Repeat your calculations for the preceding problem, given that the charge is distributed uniformly over the surface of a spherical conductor of radius 10.0 cm.

A total charge Q is distributed uniformly throughout a spherical shell of inner and outer radii $r_1\text{and}{r}_2,$ respectively. Show that the electric field due to the charge is

$\begin{array}{cccc}\overrightarrow{\text{E}}=\overrightarrow{0}\hfill & & & (r\le r_1);\hfill \\ \overrightarrow{\text{E}}=\frac{Q}{4\pi {\epsilon }_0{r}^2}\left(\frac{r^3-r_1{}^3}{r_2{}^3-r_1{}^3}\right)\widehat{\text{r}}\hfill & & & (r_1\le r\le r_2);\hfill \\ \overrightarrow{\text{E}}=\frac{Q}{4\pi {\epsilon }_0{r}^2}\widehat{\text{r}}\hfill & & & (r\ge r_2).\hfill \end{array}$

When a charge is placed on a metal sphere, it ends up in equilibrium at the outer surface. Use this information to determine the electric field of $+3.0\mu \text{C}$ charge put on a 5.0-cm aluminum spherical ball at the following two points in space: (a) a point 1.0 cm from the center of the ball (an inside point) and (b) a point 10 cm from the center of the ball (an outside point).

A large sheet of charge has a uniform charge density of $10\mu {\text{C/m}}^2\text{.}$ What is the electric field due to this charge at a point just above the surface of the sheet?

Determine if approximate cylindrical symmetry holds for the following situations. State why or why not. (a) A 300-cm long copper rod of radius 1 cm is charged with +500 nC of charge and we seek electric field at a point 5 cm from the center of the rod. (b) A 10-cm long copper rod of radius 1 cm is charged with +500 nC of charge and we seek electric field at a point 5 cm from the center of the rod. (c) A 150-cm wooden rod is glued to a 150-cm plastic rod to make a 300-cm long rod, which is then painted with a charged paint so that one obtains a uniform charge density. The radius of each rod is 1 cm, and we seek an electric field at a point that is 4 cm from the center of the rod. (d) Same rod as (c), but we seek electric field at a point that is 500 cm from the center of the rod.

A long silver rod of radius 3 cm has a charge of $\text{−}5\mu \text{C/cm}$ on its surface. (a) Find the electric field at a point 5 cm from the center of the rod (an outside point). (b) Find the electric field at a point 2 cm from the center of the rod (an inside point).

The electric field at 2 cm from the center of long copper rod of radius 1 cm has a magnitude 3 N/C and directed outward from the axis of the rod. (a) How much charge per unit length exists on the copper rod? (b) What would be the electric flux through a cube of side 5 cm situated such that the rod passes through opposite sides of the cube perpendicularly?

A long copper cylindrical shell of inner radius 2 cm and outer radius 3 cm surrounds concentrically a charged long aluminum rod of radius 1 cm with a charge density of 4 pC/m. All charges on the aluminum rod reside at its surface. The inner surface of the copper shell has exactly opposite charge to that of the aluminum rod while the outer surface of the copper shell has the same charge as the aluminum rod. Find the magnitude and direction of the electric field at points that are at the following distances from the center of the aluminum rod: (a) 0.5 cm, (b) 1.5 cm, (c) 2.5 cm, (d) 3.5 cm, and (e) 7 cm.

Charge is distributed uniformly with a density $\rho$ throughout an infinitely long cylindrical volume of radius R . Show that the field of this charge distribution is directed radially with respect to the cylinder and that

$\begin{array}{}\\ \\ E=\frac{\rho r}{2{\epsilon }_0}\hfill & & & (r\le R);\hfill \\ E=\frac{\rho R^2}{2{\epsilon }_{0}r}\hfill & & & (r\ge R).\hfill \end{array}$

Charge is distributed throughout a very long cylindrical volume of radius R such that the charge density increases with the distance r from the central axis of the cylinder according to $\rho =\alpha r,$ where $\alpha$ is a constant. Show that the field of this charge distribution is directed radially with respect to the cylinder and that

$\begin{array}{}\\ \\ E=\frac{\alpha r^2}{3{\epsilon }_0}\hfill & & & (r\le R);\hfill \\ E=\frac{\alpha R^3}{3{\epsilon }_{0}r}\hfill & & & (r\ge R).\hfill \end{array}$

The electric field 10.0 cm from the surface of a copper ball of radius 5.0 cm is directed toward the ball’s center and has magnitude $4.0\times {10}^2\text{N/C}\text{.}$ How much charge is on the surface of the ball?

Charge is distributed throughout a spherical shell of inner radius $r_1$ and outer radius $r_2$ with a volume density given by $\rho ={\rho }_0{r}_1\text{/}r,$ where ${\rho }_0$ is a constant. Determine the electric field due to this charge as a function of r , the distance from the center of the shell.

Charge is distributed throughout a spherical volume of radius R with a density $\rho =\alpha r^2,$ where $\alpha$ is a constant. Determine the electric field due to the charge at points both inside and outside the sphere.

Consider a uranium nucleus to be sphere of radius $R=7.4\times {10}^{\mathrm{-15}}\text{m}$ with a charge of 92 e distributed uniformly throughout its volume. (a) What is the electric force exerted on an electron when it is $3.0\times {10}^{\mathrm{-15}}\text{m}$ from the center of the nucleus? (b) What is the acceleration of the electron at this point?

The volume charge density of a spherical charge distribution is given by $\rho (r)={\rho }_0{e}^{\text{−}\alpha r},$ where ${\rho }_0$ and $\alpha$ are constants. What is the electric field produced by this charge distribution?