Problems

A uniform electric field of magnitude $1.1\times {10}^4\text{N/C}$ is perpendicular to a square sheet with sides 2.0 m long. What is the electric flux through the sheet?

Find the electric flux through a rectangular area $3\text{cm}\times 2\text{cm}$ between two parallel plates where there is a constant electric field of 30 N/C for the following orientations of the area: (a) parallel to the plates, (b) perpendicular to the plates, and (c) the normal to the area making a $30\text{°}$ angle with the direction of the electric field. Note that this angle can also be given as $180\text{°}+30\text{°}.$

Two large rectangular aluminum plates of area $150{\text{cm}}^2$ face each other with a separation of 3 mm between them. The plates are charged with equal amount of opposite charges, $\pm 20\mu \text{C}$. The charges on the plates face each other. Find the flux through a circle of radius 3 cm between the plates when the normal to the circle makes an angle of $5\text{°}$ with a line perpendicular to the plates. Note that this angle can also be given as $180\text{°}+5\text{°}.$

A vector field is pointed along the z-axis, $\overrightarrow{\text{v}}=\frac{\alpha }{x^2+y^2}\widehat{\text{z}}.$ (a) Find the flux of the vector field through a rectangle in the xy-plane between $a<x<b$ and $c<y<d$. (b) Do the same through a rectangle in the yz-plane between $a<z<b$ and $c<y<d$. (Leave your answer as an integral.)

Repeat the previous problem, given that the circular area is (a) in the yz-plane and (b) $45\text{°}$ above the xy-plane.

Determine the electric flux through each closed surface where the cross-section inside the surface is shown below.

A point charge q is located at the center of a cube whose sides are of length a. If there are no other charges in this system, what is the electric flux through one face of the cube?

A net flux of $1.0\times {10}^4\text{N}·{\text{m}}^2\text{/C}$ passes inward through the surface of a sphere of radius 5 cm. (a) How much charge is inside the sphere? (b) How precisely can we determine the location of the charge from this information?

The electric flux through a cubical box 8.0 cm on a side is $1.2\times {10}^3\text{N}·{\text{m}}^2\text{/C}\text{.}$ What is the total charge enclosed by the box?

A cube whose sides are of length d is placed in a uniform electric field of magnitude $E=4.0\times {10}^3\text{N/C}$ so that the field is perpendicular to two opposite faces of the cube. What is the net flux through the cube?

A total charge $5.0\times {10}^{\mathrm{-6}}\text{C}$ is distributed uniformly throughout a cubical volume whose edges are 8.0 cm long. (a) What is the charge density in the cube? (b) What is the electric flux through a cube with 12.0-cm edges that is concentric with the charge distribution? (c) Do the same calculation for cubes whose edges are 10.0 cm long and 5.0 cm long. (d) What is the electric flux through a spherical surface of radius 3.0 cm that is also concentric with the charge distribution?

Suppose that the charge density of the spherical charge distribution shown in Figure 6.23 is $\rho (r)={\rho }_{0}r\text{/}R$ for $r\le R$ and zero for $r>R.$ Obtain expressions for the electric field both inside and outside the distribution.

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.

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

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

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?

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).

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 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

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

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.

Consider a uranium nucleus to be sphere of radius $R=7.4\times {10}^{\mathrm{-15}}\text{m}$ with a charge of 92e 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?

An uncharged conductor with an internal cavity is shown in the following figure. Use the closed surface S along with Gauss’ law to show that when a charge q is placed in the cavity a total charge –q is induced on the inner surface of the conductor. What is the charge on the outer surface of the conductor?

Figure 6.46 A charge inside a cavity of a metal. Charges at the outer surface do not depend on how the charges are distributed at the inner surface since E field inside the body of the metal is zero.

A positive point charge is placed at the angle bisector of two uncharged plane conductors that make an angle of $45\text{°}.$ See below. Draw the electric field lines.

An aluminum spherical ball of radius 4 cm is charged with $5\mu \text{C}$ of charge. A copper spherical shell of inner radius 6 cm and outer radius 8 cm surrounds it. A total charge of $\mathrm{-8}\mu \text{C}$ is put on the copper shell. (a) Find the electric field at all points in space, including points inside the aluminum and copper shell when copper shell and aluminum sphere are concentric. (b) Find the electric field at all points in space, including points inside the aluminum and copper shell when the centers of copper shell and aluminum sphere are 1 cm apart.

At the surface of any conductor in electrostatic equilibrium, $E=\sigma \text{/}{\epsilon }_0.$ Show that this equation is consistent with the fact that $E=kq\text{/}{r}^2$ at the surface of a spherical conductor.

Two parallel conducting plates, each of cross-sectional area ${400\text{cm}}^2$, are 2.0 cm apart and uncharged. If $1.0\times {10}^{12}$ electrons are transferred from one plate to the other, what are (a) the charge density on each plate? (b) The electric field between the plates?

A point charge $q=\mathrm{-5.0}\times {10}^{\mathrm{-12}}\text{C}$ is placed at the center of a spherical conducting shell of inner radius 3.5 cm and outer radius 4.0 cm. The electric field just above the surface of the conductor is directed radially outward and has magnitude 8.0 N/C. (a) What is the charge density on the inner surface of the shell? (b) What is the charge density on the outer surface of the shell? (c) What is the net charge on the conductor?