Chapter 6: Gauss's LawContent check — University Physics 2, Chapter 6: Gauss's Law

What angle should there be between the electric field and the surface shown in Figure 6.11 in the previous example so that no electric flux passes through the surface?

If the electric field in Example 6.4 is $\overrightarrow{\text{E}}=mx\widehat{\text{k}},$ what is the flux through the rectangular area?

Calculate the electric flux through the closed cubical surface for each charge distribution shown in Figure 6.20.

Check that the electric fields for the sphere reduce to the correct values for a point charge.

A thin straight wire has a uniform linear charge density ${\lambda }_0.$ Find the electric field at a distance d from the wire, where d is much less than the length of the wire.

How will the system above change if there are charged objects external to the sphere?

Discuss how to orient a planar surface of area A in a uniform electric field of magnitude $E_0$ to obtain (a) the maximum flux and (b) the minimum flux through the area.

The net electric flux crossing a closed surface is always zero. True or false?

Two concentric spherical surfaces enclose a point charge q. The radius of the outer sphere is twice that of the inner one. Compare the electric fluxes crossing the two surfaces.

(a) If the electric flux through a closed surface is zero, is the electric field necessarily zero at all points on the surface? (b) What is the net charge inside the surface?

Discuss the similarities and differences between the gravitational field of a point mass m and the electric field of a point charge q.

Is the term $\overrightarrow{E}$ in Gauss’s law the electric field produced by just the charge inside the Gaussian surface?

Would Gauss’s law be helpful for determining the electric field of two equal but opposite charges a fixed distance apart?

Discuss the restrictions on the Gaussian surface used to discuss planar symmetry. For example, is its length important? Does the cross-section have to be square? Must the end faces be on opposite sides of the sheet?

Under electrostatic conditions, the excess charge on a conductor resides on its surface. Does this mean that all the conduction electrons in a conductor are on the surface?

The conductor in the preceding figure has an excess charge of $–5.0\mu \text{C}$. If a $2.0\text{-}\mu \text{C}$ point charge is placed in the cavity, what is the net charge on the surface of the cavity and on the outer surface of the conductor?

Calculate the flux through the sheet of the previous problem if the plane of the sheet is at an angle of $60\text{°}$ to the field. Find the flux for both directions of the unit normal to the sheet.

The electric flux through a square-shaped area of side 5 cm near a large charged sheet is found to be $3\times {10}^{\mathrm{-5}}\text{N}·{\text{m}}^2\text{/}\text{C}$ when the area is parallel to the plate. Find the charge density on the sheet.

A square surface of area $2{\text{cm}}^2$ is in a space of uniform electric field of magnitude ${10}^3\text{N/C}$. The amount of flux through it depends on how the square is oriented relative to the direction of the electric field. Find the electric flux through the square, when the normal to it makes the following angles with electric field: (a) $30\text{°}$, (b) $90\text{°}$, and (c) $0\text{°}$. Note that these angles can also be given as $180\text{°}+\theta$.

Consider the uniform electric field $\overrightarrow{\text{E}}=$ $(4.0\widehat{\text{j}}+3.0\widehat{\text{k}})\times {10}^3\text{N/C}\text{.}$ What is its electric flux through a circular area of radius 2.0 m that lies in the xy-plane?

An infinite charged wire with charge per unit length $\lambda$ lies along the central axis of a cylindrical surface of radius r and length l. What is the flux through the surface due to the electric field of the charged wire?

Find the electric flux through the closed surface whose cross-sections are shown below.

A point charge of $10\mu \text{C}$ is at an unspecified location inside a cube of side 2 cm. Find the net electric flux though the surfaces of the cube.

A charge q is placed at one of the corners of a cube of side a, as shown below. Find the magnitude of the electric flux through the shaded face due to q. Assume $q>0$.

The electric flux through a spherical surface is $4.0\times {10}^4\text{N}·{\text{m}}^2\text{/C}\text{.}$ What is the net charge enclosed by the surface?

Repeat the previous problem, assuming that the electric field is directed along a body diagonal of the cube.

Recall that in the example of a uniform charged sphere, ${\rho }_0=$ $Q\text{/}(\frac{4}{3}\pi R^3).$ Rewrite the answers in terms of the total charge Q on the sphere.

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?

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.

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).

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.

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?

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

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

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 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.

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?

An uncharged spherical conductor S of radius R has two spherical cavities A and B of radii a and b, respectively as shown below. Two point charges $+q_a$ and $+q_b$ are placed at the center of the two cavities by using non-conducting supports. In addition, a point charge $+q_0$ is placed outside at a distance r from the center of the sphere. (a) Draw approximate charge distributions in the metal although metal sphere has no net charge. (b) Draw electric field lines. Draw enough lines to represent all distinctly different places.

A long cylinder of copper of radius 3 cm is charged so that it has a uniform charge per unit length on its surface of 3 C/m. (a) Find the electric field inside and outside the cylinder. (b) Draw electric field lines in a plane perpendicular to the rod.

A long cylinder of aluminum of radius R meters is charged so that it has a uniform charge per unit length on its surface of $\lambda$. (a) Find the electric field inside and outside the cylinder. (b) Plot electric field as a function of distance from the center of the rod.

Two parallel plates 10 cm on a side are given equal and opposite charges of magnitude $5.0\times {10}^{\mathrm{-9}}\text{C}\text{.}$ The plates are 1.5 mm apart. What is the electric field at the center of the region between the plates?

The surface charge density on a long straight metallic pipe is $\sigma$. What is the electric field outside and inside the pipe? Assume the pipe has a diameter of 2a.

A solid cylindrical conductor of radius a is surrounded by a concentric cylindrical shell of inner radius b. The solid cylinder and the shell carry charges +Q and –Q, respectively. Assuming that the length L of both conductors is much greater than a or b, determine the electric field as a function of r, the distance from the common central axis of the cylinders, for (a) $r<a;$ (b) $a<$ $r<$ $b;$ and (c)$r>b.$

Repeat the preceding problem, with $\overrightarrow{\text{E}}=$ $2x\widehat{i}+$ $3x^2\widehat{\text{k}}.$

(a) Calculate the electric flux through the open hemispherical surface due to the electric field $\overrightarrow{\text{E}}=E_0\widehat{\text{k}}$ (see below). (b) If the hemisphere is rotated by $90\text{°}$ around the x-axis, what is the flux through it?

The electric field in a region is given by $\overrightarrow{\text{E}}=$ $a\text{/}(b+cx)\widehat{i},$ where $a=200\text{N}·\text{m/C,}$$b=2.0\text{m,}$ and $c=2.0.$ What is the net charge enclosed by the shaded volume shown below?

A fellow student calculated the flux through the square for the system in the preceding problem and got 0. What went wrong?

Two $10\text{cm}\times 10\text{cm}$ pieces of aluminum foil of thickness 0.1 mm face each other with a separation of 5 mm. One of the foils has a charge of $\text{+30}\mu \text{C}$ and the other has $\text{−}30\mu \text{C}$. (a) Find the charge density at all surfaces, i.e., on those facing each other and those facing away. (b) Find the electric field between the plates near the center assuming planar symmetry.

The infinite slab between the planes defined by $z=\text{−}a\text{/}2$ and $z=a\text{/}2$ contains a uniform volume charge density $\rho$ (see below). What is the electric field produced by this charge distribution, both inside and outside the distribution?

A non-conducting spherical shell of inner radius $a_1$ and outer radius $b_1$ is uniformly charged with charged density ${\rho }_1$ inside another non-conducting spherical shell of inner radius $a_2$ and outer radius $b_2$ that is also uniformly charged with charge density ${\rho }_2$. See below. Find the electric field at space point P at a distance r from the common center such that (a) $r>b_2,$ (b) $a_2<$ $r<$ $b_2,$ (c) $b_1<$ $r<$ $a_2,$ (d) $a_1<$ $r<$ $b_1,$ and (e) $r<a_1$.

A disk of radius R is cut in a non-conducting large plate that is uniformly charged with charge density $\sigma$ (coulomb per square meter). See below. Find the electric field at a height h above the center of the disk. $\text{(}h>>R,h<<l\text{or}w\text{).}$ (Hint: Fill the hole with $\pm \sigma .)$

Shown below are two concentric conducting spherical shells of radii $R_1$ and $R_2$, each of finite thickness much less than either radius. The inner and outer shell carry net charges $q_1$ and $q_2,$ respectively, where both $q_1$ and $q_2$ are positive. What is the electric field for (a) $r<R_1;$ (b) $R_1<$ $r<$ $R_2;$ and (c) $r>R_2?$ (d) What is the net charge on the inner surface of the inner shell, the outer surface of the inner shell, the inner surface of the outer shell, and the outer surface of the outer shell?

The Hubble Space Telescope can measure the energy flux from distant objects such as supernovae and stars. Scientists then use this data to calculate the energy emitted by that object. Choose an interstellar object which scientists have observed the flux at the Hubble with (for example, Vega³), find the distance to that object and the size of Hubble’s primary mirror, and calculate the total energy flux. (Hint: The Hubble intercepts only a small part of the total flux.)

An infinite plate sheet of charge of surface charge density $\sigma$ is shown below. What is the electric field at a distance x from the sheet? Compare the result of this calculation with that of worked out in the text.

Find the electric field of a large conducting plate containing a net charge q. Let A be area of one side of the plate and h the thickness of the plate (see below). The charge on the metal plate will distribute mostly on the two planar sides and very little on the edges if the plate is thin.