Physics for Scientists and Engineers with Modern Physics

Douglas C. Giancoli

Chapter 22

Gauss's Law - all with Video Answers

Educators

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Chapter Questions

Problem 1

(a) $31 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$;
(b) $22 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$;
(c) 0 .

Jacob Shpiece

Jacob Shpiece

Numerade Educator

Problem 2

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Problem 3

(II) A cube of side $\ell$ is placed in a uniform field $E_{0}$ with edges parallel to the field lines. ( $a$ ) What is the net flux through the cube? (b) What is the flux through each of its six faces?

Bruce Edelman

Numerade Educator

Problem 4

(II) A uniform field $\overrightarrow{\mathbf{E}}$ is parallel to the axis of a hollow hemisphere of radius $r$, Fig. $22-25 .$ ( $a$ ) What is the electric flux through the hemispherical surface? (b) What is the result if $\overrightarrow{\mathbf{E}}$ is instead perpendicular to the axis?

FIGURE $22-25$
Problem 4

Gregory Devenport

Gregory Devenport

Numerade Educator

Problem 5

(I) The total electric flux from a cubical box $28.0 \mathrm{~cm}$ on a side is $1.84 \times 10^{3} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$. What charge is enclosed by the box?

Bruce Edelman

Numerade Educator

Problem 6

(I) Figure $22-26$ shows five closed surfaces that surround various charges in a plane, as indicated. Determine the electric flux through each surface, $S_{1}, S_{2}, S_{3}, S_{4},$ and $S_{5} .$ The
surfaces are flat "pillbox" surfaces that extend only slightly above and below the plane in which the charges lie.

Jacob Shpiece

Numerade Educator

Problem 7

(II) In Fig. $22-27,$ two objects, $\mathrm{O}_{1}$ and $\mathrm{O}_{2},$ have charges $+1.0 \mu \mathrm{C}$ and $-2.0 \mu \mathrm{C}$ respectively, and a third object, $\mathrm{O}_{3},$ is electrically neutral. ( $a$ ) What is the electric flux through the surface $A_{1}$ that encloses all the three objects? (b) What is the electric flux through the surface $A_{2}$ that encloses the third object only?

Ajay Singhal

Numerade Educator

Problem 8

(II) A ring of charge with uniform charge density is completely enclosed in a hollow donut shape. An exact copy of the ring is completely enclosed in a hollow sphere. What is the ratio of the flux out of the donut shape to that out of the sphere?

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 9

(II) In a certain region of space, the electric field is constant in direction (say horizontal, in the $x$ direction), but its magnitude decreases from $E=560 \mathrm{~N} / \mathrm{C}$ at $x=0$ to $E=410 \mathrm{~N} / \mathrm{C}$ at $x=25 \mathrm{~m} .$ Determine the charge within a cubical box of side $\ell=25 \mathrm{~m},$ where the box is oriented so that four of its sides are parallel to the field lines (Fig. $22-28$ ). $x=0$
$x=25 \mathrm{~m}$

Lainey Roebuck

Numerade Educator

Problem 10

Check back soon!

Problem 11

(II) A 15.0 -cm-long uniformly charged plastic rod is sealed inside a plastic bag. The total electric flux leaving the bag is $7.3 \times 10^{5} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$. What is the linear charge density on the rod?

Bruce Edelman

Numerade Educator

Problem 12

(I) Draw the electric field lines around a negatively charged metal egg.

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 13

(I) The field just outside a 3.50-cm-radius metal ball is $6.25 \times 10^{2} \mathrm{~N} / \mathrm{C}$ and points toward the ball. What charge resides on the ball?

Jacob Shpiece

Numerade Educator

Problem 14

(I) Starting from the result of Example $22-3,$ show that the electric field just outside a uniformly charged spherical conductor is $E=\sigma / \epsilon_{0}$, consistent with Example $22-8$.

Jacob Shpiece

Numerade Educator

Problem 15

(I) A long thin wire, hundreds of meters long, carries a uniformly distributed charge of $-7.2 \mu \mathrm{C}$ per meter of length. Estimate the magnitude and direction of the electric field at points $(a) 5.0 \mathrm{~m}$ and $(b) 1.5 \mathrm{~m}$ perpendicular from the center of the wire.

Jacob Shpiece

Numerade Educator

Problem 16

(I) A metal globe has $1.50 \mathrm{mC}$ of charge put on it at the north pole. Then $-3.00 \mathrm{mC}$ of charge is applied to the south pole. Draw the field lines for this system after it has come to equilibrium.

Jacob Shpiece

Numerade Educator

Problem 17

(II) A nonconducting sphere is made of two layers. The innermost section has a radius of $6.0 \mathrm{~cm}$ and a uniform charge density of $-5.0 \mathrm{C} / \mathrm{m}^{3} .$ The outer layer has a uniform charge density of $+8.0 \mathrm{C} / \mathrm{m}^{3}$ and extends from an inner radius of $6.0 \mathrm{~cm}$ to an outer radius of $12.0 \mathrm{~cm}$. Determine the electric field for $(a) 0<r<6.0 \mathrm{~cm},(b) 6.0 \mathrm{~cm}<r<12.0 \mathrm{~cm},$ and
(c) $12.0 \mathrm{~cm}<r<50.0 \mathrm{~cm} .$ ( $d$ ) Plot the magnitude of the electric field for $0<r<50.0 \mathrm{~cm}$. Is the field continuous at the edges of the layers?

Dading Chen

Numerade Educator

Problem 18

(II) A solid metal sphere of radius $3.00 \mathrm{~m}$ carries a total charge of $-5.50 \mu \mathrm{C}$. What is the magnitude of the electric field at a distance from the sphere's center of (a) $0.250 \mathrm{~m}$, (b) $2.90 \mathrm{~m},$ (c) $3.10 \mathrm{~m},$ and $(d) 8.00 \mathrm{~m} ?$ How would the
answers differ if the sphere were $(e)$ a thin shell, or $(f)$ a solid nonconductor uniformly charged throughout?

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 19

(II) A 15.0-cm-diameter nonconducting sphere carries a total charge of $2.25 \mu \mathrm{C}$ distributed uniformly throughout its volume. Graph the electric field $E$ as a function of the distance $r$ from the center of the sphere from $r=0$ to $r=30.0 \mathrm{~cm}$

Bruce Edelman

Numerade Educator

Problem 20

(II) A flat square sheet of thin aluminum foil, $25 \mathrm{~cm}$ on a side, carries a uniformly distributed $275 \mathrm{nC}$ charge. What, approximately, is the electric field (a) $1.0 \mathrm{~cm}$ above the center of the sheet and
(b) $15 \mathrm{~m}$ above the center of the sheet?

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 21

(II) A spherical cavity of radius $4.50 \mathrm{~cm}$ is at the center of a metal sphere of radius $18.0 \mathrm{~cm}$. A point charge $Q=5.50 \mu \mathrm{C}$ rests at the very center of the cavity, whereas the metal conductor carries no net charge. Determine the electric field at a point (a) $3.00 \mathrm{~cm}$ from the center of the cavity,
(b) $6.00 \mathrm{~cm}$ from the center of the cavity,
(c) $30.0 \mathrm{~cm}$ from the center.

Dominador Tan

Numerade Educator

Problem 22

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Problem 23

(II) A solid metal cube has a spherical cavity at its center as shown in Fig. $22-29 .$ At the center of the cavity there is a point charge $Q=+8.00 \mu \mathrm{C} .$ The metal cube carries a net charge $q=-6.10 \mu \mathrm{C}$ (not including
Q). Determine $(a)$ the total charge on the surface of the spherical cavity and
(b) the total charge on the outer surface of the cube.

Bruce Edelman

Numerade Educator

Problem 24

(II) Two large, flat metal plates are separated by a distance that is very small compared to their height and width. The conductors are given equal but opposite uniform surface charge densities $\pm \sigma .$ Ignore edge effects and use Gauss's law to show $(a)$ that for points far from the edges, the electric field between the plates is $E=\sigma / \epsilon_{0}$ and
(b) that outside the plates on either side the field is zero. (c) How would your results be altered if the two plates were nonconductors?

Dominador Tan

Numerade Educator

Problem 25

(II) Suppose the two conducting plates in Problem 24 have the same sign and magnitude of charge. What then will be the electric field ( $a$ ) between them and (b) outside them on either side? (c) What if the plates are nonconducting?

Bruce Edelman

Numerade Educator

Problem 26

(II) The electric field between two square metal plates is $160 \mathrm{~N} / \mathrm{C}$. The plates are $1.0 \mathrm{~m}$ on a side and are separated by $3.0 \mathrm{~cm},$ as in Fig. $22-30 .$ What is the charge on each plate? Neglect edge effects.

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 27

(II) Two thin concentric spherical shells of radii $r_{1}$ and $r_{2}$ $\left(r_{1}<r_{2}\right)$ contain uniform surface charge densities $\sigma_{1}$ and $\sigma_{2}$, respectively (see Fig. 22-31). Determine the electric field for $(a) 0<r<r_{1},(b) r_{1}<r<r_{2},$ and
(c) $r>r_{2}$
(d) Under what conditions will $E=0$ for $r>r_{2} ?$ (e) Under what conditions will $E=0$ for $r_{1}<r<r_{2} ?$ Neglect the thickness of the shells.

Bruce Edelman

Numerade Educator

Problem 28

(II) A spherical rubber balloon carries a total charge $Q$ uniformly distributed on its surface. At $t=0$ the nonconducting balloon has radius $r_{0}$ and the balloon is then slowly blown up so that $r$ increases linearly to $2 r_{0}$ in a time $t$. Determine the electric field as a function of time $(a)$ just outside the balloon surface and $(b)$ at $r=3.2 r_{0}$.

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 29

(II) Suppose the nonconducting sphere of Example $22-4$ has a spherical cavity of radius $r_{1}$ centered at the sphere's center (Fig. 22-32). Assuming the charge $Q$ is distributed uniformly in the "shell" (between $r=r_{1}$ and $r=r_{0}$ ), determine the electric field as a function
(b) $r_{1}<r<r_{0}$
of $r$ for $(a) 0<r<r_{1}$
and $(c) r>r_{0}$

Lainey Roebuck

Numerade Educator

Problem 30

Victor Salazar

Numerade Educator

Problem 31

(II) Suppose the thick spherical shell of Problem 29 is a conductor. It carries a total net charge $Q$ and at its center there is a point charge $q$. What total charge is found on
(a) the inner surface of the shell and ( $b$ ) the outer surface of the shell? Determine the electric field for ( $c$ ) $0<r<r_{1}$,
(d) $r_{1}<r<r_{0}$, and (e) $r>r_{0}$.

Bruce Edelman

Numerade Educator

Problem 32

(II) Suppose that at the center of the cavity inside the shell (charge $Q$ ) of Fig. $22-11$ (and Example $22-3$ ), there is a point charge $q(\neq \pm Q)$. Determine the electric field for
(a) $0<r<r_{0}$, and for $(b) r>r_{0} .$ What are your answers if $(c) a=O$ and $(d) a=-O ?$

Lainey Roebuck

Numerade Educator

Problem 33

(II) A long cylindrical shell of radius $R_{0}$ and length $\ell$ $\left(R_{0} \ll \ell\right)$ possesses a uniform surface charge density (charge per unit area) $\sigma$ (Fig. $22-33$ ). Determine the electric field at points $(a)$ outside the cylinder $\left(R>R_{0}\right)$ and $(b)$ inside the cylinder $\left(0<R<R_{0}\right)$; assume the points are far from the ends and not too far from the shell $(R \ll \ell)$.
(c) Compare to the result for a long line of charge, Example $22-6 .$ Neglect $+$
the thickness of shell.

Dominador Tan

Numerade Educator

Problem 34

(II) A very long solid nonconducting cylinder of radius $R_{0}$ and length $\ell\left(R_{0} \ll \ell\right)$ possesses a uniform volume charge density $\rho_{\mathrm{E}}\left(\mathrm{C} / \mathrm{m}^{3}\right)$, Fig. $22-34 .$ Determine the electric field at points $(a)$ outside the cylinder $\left(R>R_{0}\right)$ and $(b)$ inside the cylinder $\left(R<R_{0}\right) .$ Do only for points far from the $++$
ends and for which $R \ll \ell$

Lainey Roebuck

Numerade Educator

Problem 35

(II) A thin cylindrical shell of radius $R_{1}$ is surrounded by a second concentric cylindrical shell of radius $R_{2}$ (Fig. $22-35$ ). The inner shell has a total charge $+Q$ and the outer shell $-Q .$ Assuming the length $\ell$ of the shells is much greater than $R_{1}$ or $R_{2},$ determine the electric field as a function of $R$ (the perpendicular distance from the common axis of the cylinders) for $(a) 0<R<R_{1},(b) R_{1}<R<R_{2},$ and $(c) R>R_{2}$
(d) What is the kinetic energy of an electron if it moves between $($ and concentric with) the shells in a circular orbit of radius $\left(R_{1}+R_{2}\right) / 2 ?$ Neglect thickness of shells.

Lainey Roebuck

Numerade Educator

Problem 36

(II) A thin cylindrical shell of radius $R_{1}=6.5 \mathrm{~cm}$ is surrounded by a second cylindrical shell of radius $R_{2}=9.0 \mathrm{~cm},$ as in Fig. $22-35 .$ Both cylinders are $5.0 \mathrm{~m}$ long and the inner one carries a total charge $Q_{1}=-0.88 \mu \mathrm{C}$ and the outer one $Q_{2}=+1.56 \mu \mathrm{C}$. For points far from the ends of the cylinders, determine the electric field at a radial distance $r$ from the central axis of $(a) 3.0 \mathrm{~cm},(b) 7.0 \mathrm{~cm},$ and $(c) 12.0 \mathrm{~cm} .$

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 37

(a) $1.9 \times 10^{7} \mathrm{~m} / \mathrm{s}$;
(b) $5.5 \times 10^{5} \mathrm{~m} / \mathrm{s}$.

Lainey Roebuck

Numerade Educator

Problem 38

(II) A very long solid nonconducting cylinder of radius $R_{1}$ is uniformly charged with a charge density $\rho_{\mathrm{E}}$. It is surrounded by a concentric cylindrical tube of inner radius $R_{2}$ and outer radius $R_{3}$ as shown in Fig. $22-36,$ and it too carries a uniform charge density $\rho_{\mathrm{E}}$. Determine the electric field as a function of the distance $R$ from the center of the cylinders for $(a) 0<R<R_{1},$ (b) $R_{1}<R<R_{2}$,
(c) $R_{2}<R<R_{3},$ and
(d) $R>R_{3}$.
$(e)$ If $\rho_{\mathrm{E}}=15 \mu \mathrm{C} / \mathrm{m}^{3}$
and $R_{1}=\frac{1}{2} R_{2}=\frac{1}{3} R_{3}=5.0 \mathrm{~cm}$
plot $E$ as a function of $R$ from $R=0$ to $R=20.0 \mathrm{~cm} .$ Assume
the cylinders are very long compared to $R_{3}$

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 39

(II) A nonconducting sphere of radius $r_{0}$ is uniformly charged with volume charge density $\rho_{\mathrm{E}}$. It is surrounded by a concentric metal (conducting) spherical shell of inner radius $r_{1}$ and outer radius $r_{2},$ which carries a net charge $+Q .$ Determine the resulting electric field in the regions
(a) $0<r<r_{0}$
(b) $r_{0}<r<r_{1}$
(c) $r_{1}<r<r_{2},$ and
(d) $r>r_{2}$ where the radial distance $r$ is measured from the center of the nonconducting sphere.

Bruce Edelman

Numerade Educator

Problem 40

(II) A very long solid nonconducting cylinder of radius $R_{1}$ is uniformly charged with charge density $\rho_{\mathrm{E}}$. It is surrounded by a cylindrical metal (conducting) tube of inner radius $R_{2}$ and outer radius $R_{3}$, which has no net charge (cross-sectional view shown in Fig. $22-37$ ). If the axes of the two cylinders are parallel, but displaced from each other by a distance $d$, determine the resulting electric field in the region $R>R_{3}, \quad$ where the
radial distance $R$ is measured from the metal cylinder's axis. Assume $d<\left(R_{2}-R_{1}\right)$

Lainey Roebuck

Numerade Educator

Problem 41

(II) A flat ring (inner radius $R_{0}$, outer radius $4 R_{0}$ ) is uniformly charged. In terms of the total charge $Q,$ determine the electric field on the axis at points (a) $0.25 R_{0}$ and
(b) $75 R_{0}$ from the center of the ring. [Hint: The ring can be replaced with two oppositely charged superposed disks.

Bruce Edelman

Numerade Educator

Problem 42

(II) An uncharged solid conducting sphere of radius $r_{0}$ contains two spherical cavities of radii $r_{1}$ and $r_{2},$ respectively. Point charge $Q_{1}$ is then placed within the cavity of radius $r_{1}$ and point charge $Q_{2}$ is placed within the cavity of radius $r_{2}$ (Fig. $22-38$ ). Determine the resulting electric field (magnitude and direction) at locations outside the solid sphere $\left(r>r_{0}\right)$ where $r$ is the distance from its center.

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 43

(III) A very large (i.e., assume infinite) flat slab of nonconducting material has thickness $d$ and a uniform volume charge density $+\rho_{\mathrm{E}}$.
(a) Show that a uniform electric field exists outside of this slab. Determine its magnitude $E$ and its direction (relative to the slab's surface). $(b)$ As shown in Fig. $22-39$ the slab is now aligned so that one of its surfaces lies on the line $y=x .$ At time $t=0,$ a pointlike particle (mass $m$, charge $+q$ ) is located at position $\overrightarrow{\mathbf{r}}=+y_{0} \hat{\mathbf{j}}$ and has velocity $\overrightarrow{\mathbf{v}}=v_{0} \hat{\mathbf{i}} .$ Show that the particle will collide with the slab if $v_{0} \geq \sqrt{\sqrt{2} q y_{0} \rho_{\mathrm{E}} d / \epsilon_{0} m}$
Ignore gravity.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 44

(III) Suppose the density of charge between $r_{1}$ and $r_{0}$ of the hollow sphere of Problem 29 (Fig. 22-32) varies as $\rho_{\mathrm{E}}=\rho_{0} r_{1} / r .$ Determine the electric field as a function of $r$ for $(a) 0<r<r_{1},(b) r_{1}<r<r_{0},$ and $(c) r>r_{0} .(d)$ Plot $E$ versus $r$ from $r=0$ to $r=2 r_{0}$.

Deborah Israel

Numerade Educator

Problem 45

(III) Suppose two thin flat plates measure $1.0 \mathrm{~m} \times 1.0 \mathrm{~m}$ and are separated by $5.0 \mathrm{~mm}$. They are oppositely charged with $\pm 15 \mu \mathrm{C}$. (a) Estimate the total force exerted by one plate on the other (ignore edge effects). (b) How much work would be required to move the plates from $5.0 \mathrm{~mm}$ apart to $1.00 \mathrm{~cm}$ apart?

Bruce Edelman

Numerade Educator

Problem 46

(III) A flat slab of nonconducting material (Fig. $22-40$ ) carries a uniform charge per unit volume, $\rho_{\mathrm{E}}$. The slab has thickness $d$ which is small compared to the height and breadth of the slab. Determine the electric field as a function of $x(a)$ inside the slab and ( $b$ ) outside the slab (at distances much less than the slab's height or breadth). Take the origin at the center of the slab.

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 47

(III) A flat slab of nonconducting material has thickness $2 d$, which is small compared to its height and breadth. Define the $x$ axis to be along the direction of the slab's thickness with the origin at the center of the slab (Fig. $22-41$ ). If the slab carries a volume charge density $\rho_{\mathrm{E}}(x)=-\rho_{0}$ in the region $-d \leq x<0$ and $\rho_{\mathrm{E}}(x)=+\rho_{0}$ in the region
$0<x \leq+d,$ determine the electric field $\overrightarrow{\mathbf{E}}$ as a function of $x$ in the regions $(a)$ outside the slab,
(b) $0<x \leq+d,$ and
(c) $-d \leq x<0 .$ Let $\rho_{0}$ be a positive constant.

Bruce Edelman

Numerade Educator

Problem 48

(III) An extremely long, solid nonconducting cylinder has a radius $R_{0}$. The charge density within the cylinder is a function of the distance $R$ from the axis, given by $\rho_{\mathrm{E}}(R)=\rho_{0}\left(R / R_{0}\right)^{2}$ What is the electric field everywhere inside and outside the cylinder (far away from the ends) in terms of $\rho_{0}$ and $R_{0} ?$

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 49

(III) Charge is distributed within a solid sphere of radius $r_{0}$ in such a way that the charge density is a function of the radial position within the sphere of the form: $\rho_{\mathrm{E}}(r)=\rho_{0}\left(r / r_{0}\right) .$ If the total charge within the sphere is $Q$ (and positive), what is the electric field everywhere within the sphere in terms of $Q, r_{0},$ and the radial position $r ?$

Rashmi Sinha

Numerade Educator

Problem 50

A point charge $Q$ is on the axis of a short cylinder at its center. The diameter of the cylinder is equal to its length $\ell$ (Fig. $22-42$ ). What is the total flux through the curved sides of the cylinder? [Hint: First calculate the flux through the ends. $\ell=2 R_{0}$

Lainey Roebuck

Numerade Educator

Problem 51

Write Gauss's law for the gravitational field $\overrightarrow{\mathrm{g}}$ (see Section $6-6$ ).

Jacob Shpiece

Numerade Educator

Problem 52

The Earth is surrounded by an electric field, pointing inward at every point, of magnitude $E \approx 150 \mathrm{~N} / \mathrm{C}$ near the surface. (a) What is the net charge on the Earth? (b) How many excess electrons per square meter on the Earth's surface does this correspond to?

Lainey Roebuck

Numerade Educator

Problem 53

A cube of side $\ell$ has one corner at the origin of coordinates, and extends along the positive $x, y,$ and $z$ axes. Suppose the electric field in this region is given by $\overrightarrow{\mathbf{E}}=(a y+b) \hat{\mathbf{j}}$. Determine the charge inside the cube.

Bruce Edelman

Numerade Educator

Problem 54

A solid nonconducting sphere of radius $r_{0}$ has a total charge $Q$ which is distributed according to $\rho_{\mathrm{E}}=b r,$ where $\rho_{\mathrm{E}}$ is the charge per unit volume, or charge density $\left(\mathrm{C} / \mathrm{m}^{3}\right)$ and $b$ is a constant. Determine $(a) b$ in terms of $Q,(b)$ the electric field at points inside the sphere, and $(c)$ the electric field at points outside the sphere.

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 55

A point charge of $9.20 \mathrm{nC}$ is located at the origin and a second charge of $-5.00 \mathrm{nC}$ is located on the $x$ axis at $x=2.75 \mathrm{~cm} .$ Calculate the electric flux through a sphere centered at the origin with radius $1.00 \mathrm{~m} .$ Repeat the calculation for a sphere of radius $2.00 \mathrm{~m}$.

Bruce Edelman

Numerade Educator

Problem 56

A point charge produces an electric flux of $+235 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$ through a gaussian sphere of radius $15.0 \mathrm{~cm}$ centered on the charge. ( $a$ ) What is the flux through a gaussian sphere with a radius $27.5 \mathrm{~cm} ?$ (b) What is the magnitude and sign of the charge?

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 57

A point charge $Q$ is placed a distance $r_{0} / 2$ above the surface of an imaginary spherical surface of radius $r_{0}$ (Fig. $22-43$ ).
(a) What is the electric flux through the sphere? (b) What range of values does $E$ have at the surface of the sphere?
(c) Is $\overrightarrow{\mathbf{E}}$ perpendicular to the sphere at all points? $(d)$ Is Gauss's law useful for obtaining $E$ at the surface of the sphere? \begin{tabular}{l}
4 \\
\hline
\end{tabular}

Lainey Roebuck

Numerade Educator

Problem 58

Three large but thin charged sheets are parallel to each other as shown in Fig. $22-44 .$ Sheet I has a total surface charge density of $6.5 \mathrm{nC} / \mathrm{m}^{2}$, sheet II a charge of $-2.0 \mathrm{nC} / \mathrm{m}^{2},$ and sheet III a charge of $5.0 \mathrm{nC} / \mathrm{m}^{2}$. Estimate the force per unit area on each sheet, in $\mathrm{N} / \mathrm{m}^{2} ?$

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 59

Neutral hydrogen can be modeled as a positive point charge $+1.6 \times 10^{-19} \mathrm{C}$ surrounded by a distribution of negative charge with volume density given by $\rho_{\mathrm{E}}(r)=-A e^{-2 r / a_{0}}$ where $a_{0}=0.53 \times 10^{-10} \mathrm{~m}$ is called the Bohr radius, $A$ is a constant such that the total amount of negative charge is $-1.6 \times 10^{-19} \mathrm{C},$ and $e=2.718 \cdots$ is the base of the natural log. (a) What is the net charge inside a sphere of radius $a_{0}$ ?
(b) What is the strength of the electric field at a distance $a_{0}$ from the nucleus? [Hint: Do not confuse the exponential number $e$ with the elementary charge $e$ which uses the same symbol but has a completely different meaning and value $\left(e=1.6 \times 10^{-19} \mathrm{C}\right)$

Bruce Edelman

Numerade Educator

Problem 60

A very large thin plane has uniform surface charge density $\sigma$. Touching it on the right (Fig. $22-45$ ) is a long wide slab of thickness $d$ with uniform volume charge density $\rho_{\mathrm{E}}$. Determine the electric field
(a) to the left of the plane, $(b)$ to the right of the slab, and ( $c$ ) everywhere inside the slab.

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 61

A sphere of radius $r_{0}$ carries a volume charge density $\rho_{\mathrm{E}}$ (Fig. 22-46). A spherical cavity of radius $r_{0} / 2$ is then scooped out and left empty, as shown. (a) What is the magnitude and direction of the electric field at point A?
(b) What is the direction and magnitude of the electric field at point $\mathrm{B}$ ? Points $\mathrm{A}$ and $\mathrm{C}$ are at the centers of the respective spheres.

Bruce Edelman

Numerade Educator

Problem 62

Check back soon!

Problem 63

Three very large sheets are separated by equal distances of $15.0 \mathrm{~cm}$ (Fig. $22-47$ ). The first and third sheets are very thin and nonconducting and have charge per unit area $\sigma$ of $+5.00 \mu \mathrm{C} / \mathrm{m}^{2}$ and $-5.00 \mu \mathrm{C} / \mathrm{m}^{2}$ respectively. The
middle sheet is conducting but has no net charge.
(a) What is the electric field inside the middle sheet? What is the electric field $(b)$ between the left and middle sheets, and (c) between the middle and right sheets?
(d) What is the charge density on the surface of the middle sheet facing the left sheet, and $(e)$ on the surface facing the right sheet?

Bruce Edelman

Numerade Educator

Problem 64

In a cubical volume, $0.70 \mathrm{~m}$ on a side, the electric field is
$$
\overrightarrow{\mathbf{E}}=E_{0}\left(1+\frac{z}{a}\right) \hat{\mathbf{i}}+E_{0}\left(\frac{z}{a}\right) \hat{\mathbf{j}}
$$
where $E_{0}=0.125 \mathrm{~N} / \mathrm{C}$ and $a=0.70 \mathrm{~m} .$ The cube has its sides parallel to the coordinate axes, Fig. 22-48. Determine the net charge within the cube.

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 65

A conducting spherical shell (Fig. 22-49) has inner radius $=10.0 \mathrm{~cm},$ outer radius $=15.0 \mathrm{~cm},$ and has a $+3.0 \mu \mathrm{C}$ point charge at the center. A charge of $-3.0 \mu \mathrm{C}$ is put on the conductor. ( $a$ ) Where on the conductor does the $-3.0 \mu \mathrm{C}$ end up? $(b)$ What is the electric field both inside and outside the shell?

Bruce Edelman

Numerade Educator

Problem 66

A hemisphere of radius $R$ is placed in a charge-free region of space where a uniform electric field exists of magnitude $E$ directed perpendicular to the hemisphere's circular base (Fig. $22-50$ ). (a) Using the definition of $\Phi_{E}$ through an "open" surface, calculate (via explicit integration) the electric flux through the hemisphere. [Hint: In Fig. $22-50$ you can see that, on the surface of a sphere, the infinitesimal area located between the angles $\theta$ and $\theta+d \theta$ is $\left.d A=(2 \pi R \sin \theta)(R d \theta)=2 \pi R^{2} \sin \theta d \theta .\right] \quad(b) \quad$ Choose $\quad$ an
appropriate gaussian surface and use Gauss's law to much more easily obtain the same result $d A=2 \pi R^{2} \sin \theta d \theta$
for the electric flux through the hemisphere.

Lainey Roebuck

Numerade Educator

Problem 67

(III) An electric field is given by
$$
\mathbf{E}=E_{x 0} e^{-\left(\frac{x+y}{a}\right)^{2}} \hat{\mathbf{i}}+E_{y 0} e^{-\left(\frac{x+y}{a}\right)^{2}} \hat{\mathbf{j}}
$$
where $E_{x 0}=50 \mathrm{~N} / \mathrm{C}, E_{y 0}=25 \mathrm{~N} / \mathrm{C},$ and $a=1.0 \mathrm{~m} .$ Given
a cube with sides parallel to the coordinate axes, with one corner at the origin (as in Fig. $22-48$ ), and with sides of length $1.0 \mathrm{~m},$ estimate the flux out of the cube using a spreadsheet or other numerical method. How much total charge is enclosed by the cube?

Lainey Roebuck

Numerade Educator