Samuel J. Ling; William Moebs; Jeff Sanny

Additional Problems

72.

A vector field $\vec{E}$ (not necessarily an electric field; note units) is given by $\vec{E} = 3 x^{2} \hat{k} .$ Calculate $\int_{S} \vec{E} \cdot \hat{n} d a,$ where S is the area shown below. Assume that $\hat{n} = \hat{k} .$

A square S with length of each side equal to a is shown in the xy plane.

73.

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

74.

A circular area S is concentric with the origin, has radius a, and lies in the yz-plane. Calculate $\int_{S} \vec{E} \cdot \hat{n} d A$ for $\vec{E} = 3 z^{2} \hat{i} .$

75.

(a) Calculate the electric flux through the open hemispherical surface due to the electric field $\vec{E} = E_{0} \hat{k}$ (see below). (b) If the hemisphere is rotated by $90 °$ around the x-axis, what is the flux through it?

A hemisphere with radius R is shown with its base in the xy plane and center of base at the origin. An arrow is shown beside it, labeled vector E equal to E0 k hat.

76.

Suppose that the electric field of an isolated point charge were proportional to $1 / r^{2 + σ}$ rather than $1 / r^{2} .$ Determine the flux that passes through the surface of a sphere of radius R centered at the charge. Would Gauss’s law remain valid?

77.

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

Figure shows a cuboid with one corner on the origin of the coordinate axes. Its length along the x axis is 2 m, along y axis is 1.5 m and along z axis is 1 m. An arrow outside the cuboid points along the x axis. It is labeled vector E.

78.

Two equal and opposite charges of magnitude Q are located on the x-axis at the points +a and –a, as shown below. What is the net flux due to these charges through a square surface of side 2a that lies in the yz-plane and is centered at the origin? (Hint: Determine the flux due to each charge separately, then use the principle of superposition. You may be able to make a symmetry argument.)

A shaded square is shown in the yz plane with its center at the origin. Its side parallel to z axis is labeled to be of length 2a. A charge labeled plus Q is shown on the positive x axis at a distance a from the origin. A charge labeled minus Q is shown on the negative x axis at a distance a from the origin.

79.

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

80.

A $10 cm \times 10 cm$ piece of aluminum foil of 0.1 mm thickness has a charge of $20 μ C$ that spreads on both wide side surfaces evenly. You may ignore the charges on the thin sides of the edges. (a) Find the charge density. (b) Find the electric field 1 cm from the center, assuming approximate planar symmetry.

81.

Two $10 cm \times 10 cm$ 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 $+30 μ C$ and the other has $− 30 μ C$ . (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.

82.

Two large copper plates facing each other have charge densities $\pm 4.0 {C/m}^{2}$ on the surface facing the other plate, and zero in between the plates. Find the electric flux through a $3 cm \times 4 cm$ rectangular area between the plates, as shown below, for the following orientations of the area. (a) If the area is parallel to the plates, and (b) if the area is tilted $θ = 30 °$ from the parallel direction. Note, this angle can also be $θ = 180 ° + 30 ° .$

Figure shows two parallel plates and a dotted line exactly between the two, parallel to them. A third plate forms an angle theta with the dotted line.

83.

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

Figure shows a cuboid with its center at the origin of the coordinate axes. Arrows perpendicular to the surfaces of the cuboid point outward. The arrows along positive x and y axes are labeled infinity and the arrows along the negative x and y axes are labeled minus infinity. The cuboid is labeled rho. Its top surface is labeled z equal to plus a by 2 and its bottom surface is labeled z equal to minus a by 2.

84.

A total charge Q is distributed uniformly throughout a spherical volume that is centered at $O_{1}$ and has a radius R. Without disturbing the charge remaining, charge is removed from the spherical volume that is centered at $O_{2}$ (see below). Show that the electric field everywhere in the empty region is given by

$\vec{E} = \frac{Q \vec{r}}{4 π ε_{0} R^{3}},$

where $\vec{r}$ is the displacement vector directed from $O_{1} to O_{2} .$

Figure shows a circle with center O1 and radius R. Another smaller circle with center O2 is shown within it. An arrow from O1 to O2 is labeled vector r.

85.

A non-conducting spherical shell of inner radius $a_{1}$ and outer radius $b_{1}$ is uniformly charged with charged density $ρ_{1}$ inside another non-conducting spherical shell of inner radius $a_{2}$ and outer radius $b_{2}$ that is also uniformly charged with charge density $ρ_{2}$ . 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) $r < a_{1}$ .

Figure shows two concentric circular shells. The inner and outer radii of the inner shell are a1 and a2 respectively. The inner and outer radii of the outer shell are a2 and b2 respectively. The distance from the center to a point P between the two shells is labeled r.

86.

Two non-conducting spheres of radii $R_{1}$ and $R_{2}$ are uniformly charged with charge densities $ρ_{1}$ and $ρ_{2},$ respectively. They are separated at center-to-center distance a (see below). Find the electric field at point P located at a distance r from the center of sphere 1 and is in the direction $θ$ from the line joining the two spheres assuming their charge densities are not affected by the presence of the other sphere. (Hint: Work one sphere at a time and use the superposition principle.)

Two circles are shown side by side with the distance between their centers being a. The bigger circle has radius R1 and the smaller one has radius R2. An arrow r is shown from the center of the bigger circle to a point P outside the circles. r forms an angle theta with a.

87.

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

A plate with length l and width w has a hole in the center. A point P above the plate is at a distance h from its center.

88.

Concentric conducting spherical shells carry charges Q and –Q, respectively (see below). The inner shell has negligible thickness. Determine the electric field for (a) $r < a;$ (b) $a < r < b;$ (c) $b < r < c;$ and (d) $r > c .$

Section of two concentric spherical shells is shown. The inner shell has a radius a. It is labeled Q and has plus signs around it. The outer shell has an inner radius b and an outer radius c. It is labeled minus Q and has minus signs around it.

89.

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?

Figure shows section of two concentric spherical shells. The inner one has radius R1 and the outer one has radius R2.

90.

A point charge of $q = 5.0 \times 10^{−8} C$ is placed at the center of an uncharged spherical conducting shell of inner radius 6.0 cm and outer radius 9.0 cm. Find the electric field at (a) $r = 4.0 cm$ , (b) $r = 8.0 cm$ , and (c) $r = 12.0 cm$ . (d) What are the charges induced on the inner and outer surfaces of the shell?