Understanding Physics

Karen Cummings, Priscilla W. Laws, Edward F. Redish

Chapter 24

Gauss’ Law - all with Video Answers

Educators

Chapter Questions

Problem 1

Cube The cube in Fig. $24-25$ has edge lengths of $1.40 \mathrm{~m}$ and is oriented as shown with its bottom face in the $x-y$ plane at $z=$ $0.00 \mathrm{~m}$. Find the electric flux through the right face if the uniform electric field, in newtons per coulomb, is given by (a) $6.00 \hat{i}$,
(b) $-2.00 \hat{j}$, and
(c) $-3.00 \hat{\mathrm{i}}+$
$4.00 \hat{k}$. (d) What is the total flux through the cube for each of these fields?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 2

Square Surface The square and 10 . surface shown in Fig. 24-26 measures $3.2 \mathrm{~mm}$ on each side. It is immersed in a uniform electric field with magnitude $|\vec{E}|=1800 \mathrm{~N} / \mathrm{C}$. The field lines make an angle of $35^{\circ}$ with a normal to the surface, as shown. Take that normal to be directed "outward," as though the surface were one face of a box. Calculate the electric flux through the surface.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 3

Charge at Center of Cube A point charge of $1.8 \mu \mathrm{C}$ is at the center of a cubical Gaussian surface $55 \mathrm{~cm}$ on edge. What is the net electric flux through the surface?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 4

Four Charges You have four point charges, $2 q, q,-q$, and $-2 q .$ If possible describe how you would place a closed surface that encloses at least the charge $2 q$ (and perhaps other charges) and through which the net electric flux is (a) 0 (b) $+3 q / \varepsilon_{0}$ and (c) $-2 q / \varepsilon_{0}$ -

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 5

Flux Through Cube Find the net flux through the cube of Problem 1 and Fig. $24-25$ if the electric field is given by (a) $\vec{E}=(3.00 y[\mathrm{~N} /(\mathrm{C} \cdot \mathrm{m})]) \hat{\mathrm{j}}$
and (b) $\vec{E}=-(4.00 \mathrm{~N} / \mathrm{C}) \mathrm{i}+(6.00 \mathrm{~N} / \mathrm{C}+3.00 y[\mathrm{~N} /(\mathrm{C} \cdot \mathrm{m})] \hat{\mathrm{j}}$. (c) In each
case, how much charge is enclosed by the cube?

Yaqub Khan

Numerade Educator

Problem 6

Butterfly Net In Fig. 24-27, a butterfly net is in a uniform electric field of magnitude $\vec{E}$. The rim, a circle of radius $a$, is aligned perpendicular to the field. Find the electric flux through the netting.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 7

Earth's Atmosphere It is found experimentally that the electric field in a certain region of
Earth's atmosphere is directed vertically down. At an altitude of $300 \mathrm{~m}$ the field has magnitude $60.0 \mathrm{~N} / \mathrm{C}$, at an altitude of $200 \mathrm{~m}$, the magnitude is $100 \mathrm{~N} / \mathrm{C}$. Find the net amount of charge contained in a cube $100 \mathrm{~m}$ on edge, with horizontal faces at altitudes of 200 and $300 \mathrm{~m}$. Neglect the curvature of Earth.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 8

Shower When a shower is turned on in a closed bathroom, the splashing of the water on the bare tub can fill the room's air with negatively charged ions and produce an electric field in the air as great as $1000 \mathrm{~N} / \mathrm{C}$. Consider a bathroom with dimensions of $2.5 \mathrm{~m} \times 3.0 \mathrm{~m} \times 2.0 \mathrm{~m}$. Along the ceiling, floor, and four walls, approximate the electric field in the air as being directed perpendicular to the surface and as having a uniform magnitude of $600 \mathrm{~N} / \mathrm{C}$. Also, treat those surfaces as forming a closed Gaussian surface around the room's air. What are (a) the volume charge density $\rho$ and (b) the number of excess elementary charges $e$ per cubic meter in the room's air?

Keshav Singh

Numerade Educator

Problem 9

Point Charge A point charge $q$ is placed at one corner of a cube of edge $a$. What is the flux through each of the cube faces? (Hint:
Use Gauss' law and symmetry arguments.)

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 10

Surface of Cube At each point on the surface of the cube shown in Fig $24-25$, the electric field is along the $y$ -axis. The length of each edge of the cube is $3.0 \mathrm{~m}$. On the right surface of the cube. $\vec{E}=(-34 \mathrm{~N} / \mathrm{C}) \hat{\mathrm{j}}$, and on the left face of the cube $\vec{E}=(+20 \mathrm{~N} / \mathrm{C}) \hat{\mathrm{j}}$
Determine the net charge contained within the cube.

Raj Bala

Numerade Educator

Problem 11

Conducting Sphere A conducting sphere of radius $10 \mathrm{~cm}$ has an unknown charge. If the electric field $15 \mathrm{~cm}$ from the center of the sphere has the magnitude $3.0 \times 10^{3} \mathrm{~N} / \mathrm{C}$ and is directed radially inward, what is the net charge on the sphere?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 12

Charge Causes Flux A point charge causes an electric flux of $-750 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$ to pass through a spherical Gaussian surface of $10.0 \mathrm{~cm}$ radius centered on the charge. (a) If the radius of the Gaussian surface were doubled, how much flux would pass through the surface? (b) What is the value of the point charge?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 13

Rutherford In a 1911 paper, Ernest Rutherford said: "In order to form some idea of the forces required to deflect an $\alpha$ particle through a large angle, consider an atom [as] containing a point positive charge $Z e$ at its center and surrounded by a distribution of negative electricity $-Z e$ uniformly distributed within a sphere of radius $R$. The electric field $E \ldots$ at a distance $r$ from the center for a point inside the atom [is]
$$
E=\frac{Z e}{4 \pi \varepsilon_{0}}\left(\frac{1}{r^{2}}-\frac{r}{R^{3}}\right)^{\prime}
$$
Verify this equation.

Keshav Singh

Numerade Educator

Problem 14

Concentric Spheres Two charged concentric spheres have radii of $10.0 \mathrm{~cm}$ and $15.0 \mathrm{~cm}$. The charge on the inner sphere is $4.00 \times$ $10^{-8} \mathrm{C}$, and that on the outer sphere is $2.00 \times 10^{-8} \mathrm{C}$. Find the electric field (a) at $r=12.0 \mathrm{~cm}$ and (b) at $r=20.0 \mathrm{~cm}$.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 15

Proton A proton with speed $v=3.00 \times 10^{5} \mathrm{~m} / \mathrm{s}$ orbits just outside a charged sphere of radius $r=1.00 \mathrm{~cm} .$ What is the charge on the sphere?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 16

Charge at Center of Shell A point charge $+q$ is placed at the center of an electrically neutral, spherical conducting shell with inner radius $a$ and outer radius $b .$ What charge appears on (a) the inner surface of the shell and (b) the outer surface? What is the net electric field at a distance $r$ from the center of the shell if (c) $r<a$,
(d) $b>r>a$, and (e) $r>b$ ? Sketch field lines for those three regions. For $r>b$, what is the net electric field due to (f) the central point charge plus the inner surface charge and (g) the outer surface charge? A point charge $-q$ is now placed outside the shell. Does this point charge change the charge distribution on (h) the outer surface and (i) the inner surface? Sketch the field lines now. (j) Is there an electrostatic force on the second point charge? (k) Is there a net electrostatic force on the first point charge? (1) Does this situation violate Newton's Third Law?

Surjit Tewari

Numerade Educator

Problem 17

Solid Nonconducting Sphere A solid nonconducting sphere of radius $R$ has a nonuniform charge distribution of volume charge density $\rho=\rho_{s} r / R$, where $\rho_{s}$ is a constant and $r$ is the distance from the center of the sphere. Show (a) that the total charge on the sphere is $Q=\pi \rho_{s} R^{3}$ and $(b)$ that
$$
|\vec{E}|=k \frac{|Q|}{R^{4}} r^{2}
$$
gives the magnitude of the electric field inside the sphere.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 18

Hydrogen Atom A hydrogen atom can be considered as having a central point-like proton of positive charge $+e$ and an electron of negative charge $-e$ that is distributed about the proton according to the volume charge density $\rho=A \exp \left(-2 r / a_{1}\right) .$ Here $A$ is a constant, $a_{1}=0.53 \times 10^{-10} \mathrm{~m}$ is the Bohr radius, and $r$ is the distance from the center of the atom. (a) Using the fact that hydrogen is electrically neutral, find $A .$ (b) Then find the electric field produced by the atom at the Bohr radius.

Suzanne W.

Numerade Educator

Problem 19

Sphere of Radius $a$ In Fig $24-28$ an insulating sphere, of radius $a$ and charge $+q$ uniformly distributed throughout its volume, is concentric with a spherical conducting shell of inner radius $b$ and outer radius $c$. This shell has a net charge of $-q$. Find expressions for the electric field, as a function of the radius $r,(a)$ within the sphere $(r<a)$, (b) between the sphere and the shell $(a<r<b)$, (c) inside the shell
$(b<r<c)$, and $(\mathrm{d})$ outside the shell $(r>c) .(\mathrm{e}) \mathrm{What}$ are the charges on the inner and outer surfaces of the shell?

Emily Anderson

Numerade Educator

Problem 20

Uniform Volume Charge Density Figure 24-29a shows a spherical shell of charge with uniform volume charge density $\rho .$ Plot $E$ due to the shell for distances $r$ from the center of the shell ranging from zero to $30 \mathrm{~cm}$. Assume that $\rho=1.0 \times 10^{-6} \mathrm{C} / \mathrm{m}^{3}, a=10 \mathrm{~cm}$,
and $b=20 \mathrm{~cm}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 21

Nonconducting Spherical Shell In Fig. $24-29 b$, a nonconducting spherical shell, of inner radius $a$ and outer radius $b$, has a positive volume charge density $\rho=A / r$ (within its thickness), where $A$ is a constant and $r$ is the distance from the center of the shell. In addition, a positive point charge $q$ is located at that center. What value should $A$ have if the electric field in the shell $(a \leq r \leq b)$ is to be uniform? (Hint: The constant $A$ depends on $a$ but not on $b .$ )

Keshav Singh

Numerade Educator

Problem 22

Show That A nonconducting sphere has a uniform volume charge density $\rho$. Let $\vec{r}$ be the vector from the center of the sphere to a general point $P$ within the sphere. (a) Show that the electric field at $P$ is given by $\vec{E}=\rho \vec{r}$ $13 \varepsilon_{0}$. (Note that the result is independent of the radius of the sphere.) (b) A spherical cavity is hollowed out of the sphere, as shown in Fig. 24-30. Using superposition concepts, show that the electric field at all points within
the cavity is uniform and equal to $\bar{E}=\rho \vec{a} / 3 \varepsilon_{0}$, where $\vec{a}$ is the position vector from the center of the sphere to the center of the cavity. (Note that this result is independent of the radius of the sphere and the radius of the cavity.)

Keshav Singh

Numerade Educator

Problem 23

Spherically Symmetrical A spherically symmetrical but nonuniform volume distribution of charge produces an electric field of magnitude $|\vec{E}|=K r^{4}$, directed radially outward from the center of the sphere. Here $r$ is the radial distance from that center, and $K$ is a positive constant. What is the volume density $\rho$ of the charge distribution as a function of $r$ ?

Katie Mcalpine

Numerade Educator

Problem 24

Long Metal Tube Figure $24-31$ shows a section of a long, thinwalled metal tube of radius $R$, with a positive charge per unit length $\lambda$ on its surface. Derive expressions for $|\vec{E}|$ in terms of the distance $r$ from the tube axis, considering both (a) $r>R$ and (b) $r<R$. Plot your results for the range $r=0$ to $r=5.0 \mathrm{~cm}$, assuming that $\lambda=2.0 \times$
$10^{-8} \mathrm{C} / \mathrm{m}$ and $R=3.0 \mathrm{~cm}$. (Hint:
Use cylinderical Gaussian surfaces, coaxial with the metal tube.)

Anand Jangid

Numerade Educator

Problem 25

Infinite Line of Charge An infinite line of charge produces a field magnitude of $4.5 \times 10^{4} \mathrm{~N} / \mathrm{C}$ at a distance of $2.0 \mathrm{~m} .$ Calculate the amount of linear charge density $|\lambda|$.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 26

Long Straight Wire A long, straight wire has fixed negative charge with a linear charge density of $-3.6 \mathrm{nC} / \mathrm{m}$. The wire is to be
enclosed by a thin, nonconducting cylinder of outside radius $1.5 \mathrm{~cm}$, coaxial with the wire. The cylinder is to have positive charge on its outside surface with a surface charge density $\sigma$ such that the net external electric field is zero. Calculate the required $\sigma$.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 27

Cylindrical Rod A very long conducting cylindrical rod of length $L$ with a total charge $+q$ is surrounded by a conducting cylindrical shell (also of length $L$ ) with total charge $-2 q$, as shown in Fig. $24-32$. Use Gauss' law to find (a) the electric field at points outside the conducting shell, (b) the distribution of charge on the shell, and (c) the electric field in the region between the shell and rod. Neglect end effects.

Bruce Edelman

Numerade Educator

Problem 28

Solid Cylinder A long, nonconducting, solid cylinder of radius $4.0$
$\mathrm{cm}$ has a nonuniform volume charge density $\rho$ that is a function of the radial distance $r$ from the axis of the cylinder, as given by $\rho=$ $A r^{2}$ with $A=2.5 \mu \mathrm{C} / \mathrm{m}^{5} .$ What is the magnitude of the electric field at a radial distance of (a) $3.0 \mathrm{~cm}$ and (b) $5.0 \mathrm{~cm}$ from the axis of the cylinder?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 29

Two Concentric Cylinders Two long, charged, concentric cylinders have radii of $3.0$ and $6.0 \mathrm{~cm}$. Assume the outer cylinder is hollow. The charge per unit length is $5.0 \times 10^{-6} \mathrm{C} / \mathrm{m}$ on the inner cylinder and $-7.0 \times 10^{-6} \mathrm{C} / \mathrm{m}$ on the outer cylinder. Find the electric field at (a) $r=4.0 \mathrm{~cm}$ and (b) $r=8.0 \mathrm{~cm}$, where $r$ is the radial distance from the common central axis.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 30

Geiger Counter Figure $24-33$ shows a Geiger counter, a device used to detect ionizing radiation (radiation that causes ionization of atoms). The counter consists of a thin, positively charged central wire surrounded by a concentric, circular, conducting cylinder with an equal negative charge. Thus, a strong radial electric field is set up inside the cylinder. The cylinder contains a low-pressure inert gas. When a particle of radiation enters the device through the cylinder wall, it ionizes a few of the gas atoms The resulting free electrons (labelled e) are drawn to the positive wire. However, the electric field is so intense that, between
collisions with other gas atoms, the free electrons gain energy sufficient to ionize these atoms also. More free electrons are thereby created, and the process is repeated until the electrons reach the wire. The resulting "avalanche" of electrons is collected by the wire generating a signal that is used to record the passage of the original particle of radiation. Suppose that the radius of the central wire is $25 \mu \mathrm{m}$, the radius of the cylinder $1.4 \mathrm{~cm}$, and the length of the tube $16 \mathrm{~cm}$. If the electric field component $E_{r}$ at the cylinder's inner wall is $+2.9 \times$ $10^{4} \mathrm{~N} / \mathrm{C}$, what is the total positive charge on the central wire?

Keshav Singh

Numerade Educator

Problem 31

Charge Is Distributed Uniformly Charge is distributed uniformly throughout the volume of an infinitely long cylinder of
radius $R$. (a) Show that, at a distance $r$ from the cylinder axis (for $r<R)$
$$
|\vec{E}|=\frac{|\rho| r}{2 \varepsilon_{0}}
$$
where $|\rho|$ is the amount of volume charge density. (b) Write an expression for $|\vec{E}|$ when $r>R$

Keshav Singh

Numerade Educator

Problem 32

Parallel } & \text { Sheets Figure } & 24-34\end{array}$ shows cross sections through two large, parallel, nonconducting sheets with identical distributions of positive charge with area charge density $\sigma$. What is $\vec{E}$ at points (a) above the sheets, (b) between them, and (c) below them?

Keshav Singh

Numerade Educator

Problem 33

Square Metal Plate A square metal plate of edge length $8.0 \mathrm{~cm}$ and negligible thickness has a total charge of $6.0 \times 10^{-6} \mathrm{C}$ (a) Estimate the magnitude $E$ of the electric field just off the center of the plate (at, say, a distance of $0.50 \mathrm{~mm}$ ) by assuming that the charge is spread uniformly over the two faces of the plate. (b) Estimate $E$ at a distance of $30 \mathrm{~m}$ (large relative to the plate size) by assuming that the plate is a point charge.

Keshav Singh

Numerade Educator

Problem 34

Thin Metal Plates Two large, thin metal plates are parallel and close to each other. On their inner faces, the plates have excess surface charge of opposite signs. The amount of charge per unit area is given by $|\sigma|=7.0 \times 10^{-22} \mathrm{C} / \mathrm{m}^{2}$, with the negatively charged plate on the left. What are the magnitude and direction of the electric field $\vec{E}$ (a) to the left of the plates, (b) to the right of the plates, and (c) between the plates?

Keshav Singh

Numerade Educator

Problem 35

Ball on Thread In Fig. 24-35, a small, nonconducting ball of mass $m=1.0 \mathrm{mg}$ and charge $q=2.0 \times 10^{-8} \mathrm{C}$ (distributed uniformly through its volume) hangs from an insulating thread that makes an angle $\theta$ $=30^{\circ}$ with a vertical, uniformly charged nonconducting sheet (shown in cross section). Considering the gravitational force on the ball and assuming that the sheet extends far vertically and into and out of the page, calculate the surface charge density $\sigma$ of the sheet.

Raj Bala

Numerade Educator

Problem 36

Large Metal Plates Two large metal plates of area $1.0 \mathrm{~m}^{2}$ face each other. They are $5.0 \mathrm{~cm}$ apart and have equal but opposite charges on their inner surfaces. If the magnitude $|\vec{E}|$ of the electric field between the plates is $55 \mathrm{~N} / \mathrm{C}$, what is the amount of charge on each plate? Neglect edge effects.

Chad Smith

Numerade Educator

Problem 37

An Electron Is Shot An electron is shot directly toward the center of a large metal plate that has excess negative charge with surface charge density $-2.0 \times 10^{-6} \mathrm{C} / \mathrm{m}^{2} .$ If the initial kinetic energy of the electron is $1.60 \times 10^{-17} \mathrm{~J}$ and if the electron is to stop (owing to electrostatic repulsion from the plate) just as it reaches the plate, how far from the plate must it be shot?

Keshav Singh

Numerade Educator

Problem 38

Planar Slab A planar slab of thickness $d$ has a uniform volume charge density $\rho$. Find the magnitude of the electric field at all points in space both (a) within and (b) outside the slab, in terms of $x$, the distance measured from the central plane of the slab.

Katie Mcalpine

Numerade Educator

Problem 39

Photocopying Machine The electric field just above the surface of the charged drum of a photocopying machine has a magnitude $|\vec{E}|$ of $2.3 \times 10^{5} \mathrm{~N} / \mathrm{C}$. What is the surface charge density on the drum, assuming that the drum is a conductor?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 40

Space Vehicles Space vehicles traveling through Earth's radiation belts can intercept a significant number of electrons. The resulting charge buildup can damage electronic components and disrupt operations. Suppose a spherical metallic satellite $1.3 \mathrm{~m}$ in diameter accumulates $-2.4 \mu \mathrm{C}$ of charge in one orbital revolution. (a) Find the resulting surface charge density. (b) Calculate the magnitude of the electric field just outside the surface of the satellite due to the surface charge.

Raj Bala

Numerade Educator

Problem 41

Charged Sphere A uniformly charged conducting sphere of $1.2 \mathrm{~m}$ diameter has a surface charge density of $8.1 \mu \mathrm{C} / \mathrm{m}^{2} .$ (a) Find the net charge on the sphere. (b) What is the total electric flux leaving the surface of the sphere?

Bettina Hanlon

Numerade Educator

Problem 42

Arbitrary Shape Conductor An isolated conductor of arbitrary shape has a net charge of $+10 \times 10^{-6} \mathrm{C}$. Inside the conductor is a cavity within which is a point charge $q=+3.0 \times 10^{-6} \mathrm{C}$. What is the charge (a) on the cavity wall and (b) on the outer surface of the conductor?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 43

If/Can If the electric field in a region of space is zero, can you conclude there are no electric charges in that region? Explain.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 44

If/Than If there are fewer electric field lines leaving a Gaussian surface than there are entering the surface, what can you conclude about the net charge enclosed by that surface?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 45

Net Flux What is the net electric flux through each of the closed surfaces in Fig. 24-36 if the value of $q$ is $+1.6 \times 10^{-19} \mathrm{C}$ ?

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 46

Net Flux Two What is the net electric flux through each of the closed surfaces in Fig. 24-37 if the value of $q$ is $8.85 \times 10^{-12} \mathrm{C}$ ? Explain the reasons for your answers.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 47

Fair Weather During fair weather, an electric field of about 100 $\mathrm{N} / \mathrm{C}$ points vertically downward into Earth's atmosphere. Assuming that this field arises from charge distributed in a spherically symmetric manner over the surface of Earth, determine the net charge of Earth and its atmosphere if the radius of Earth and its atmosphere is $6.37 \times 10^{6} \mathrm{~m}$.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 48

Hollow Sphere Suppose a charge is located at the center of a hollow sphere as shown in Fig. $24-38$.
(a) Are the intersections of the field lines with the surface of the sphere uniformly distributed throughout? In other
words, is the density of lines passing through the surface of the sphere uniform? Explain why or why not.
(b) Consider surface elements $A$ and $B$, which have exactly the same area. Is the number of field lines passing through surface element $A$ greater than, less than, or equal to the number of field lines through surface element $B ?$ Explain.
(c) Is the flux through surface element $A$ greater than, less than, or equal to the flux through surface element $B$ ? Explain.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 49

Center of Cube Suppose a charge is located at the center of the cube shown in Fig. 24-39.
(a) Are the intersections of the field lines with a side of the cube uniformly distributed across the side? In other words, is the density of lines passing through the box uniform? Explain why or why not.
(b) Is the number of field lines through surface element $A$ greater than, less than, or equal to the number of field lines through surface element $B ?$ Explain.
(c) Is the flux through surface element $A$ greater than, less than, or equal to the flux through surface element $B ?$ Explain.

Manish Kumar ( Iit K )

Manish Kumar ( Iit K )

Numerade Educator

Problem 50

Using Gaus' Law Gauss' law is usually written as an equation in the form
$$
\oint \vec{E} \cdot d \vec{A}=q^{\mathrm{enc}} / \varepsilon_{0}
$$
(a) For this equation, specify what each term in this equation means and how it is to be calculated when doing some specific (but arbitrary - not a special case) calculation.
A long thin cylindrical shell like that shown in Fig. $24-40$ has length $L$ and radius $R$ with $L \gg R$ and is uniformly covered with a charge $Q$. If we look for the field near the cylinder some-
where about the middle, we can treat the cylinder as if it were an infinitely long cylinder. Using this assumption, we can calculate the magnitude and direction of the field at a point a distance $d$ from the axis of the cylinder (outside the cylindrical shell; i.e., $L>d>R$ but $d$ not very close to $R$ ) using Gauss's law. Do so by explicitly following the steps below.
(b) Select an appropriate Gaussian surface. Explain why you chose it.
(c) Carry out the integral on the left side of the equation, expressing it in terms of the unknown value of the magnitude of the $E$ field.
(d) What is the relevant value of $q$ for your surface?
(e) Use your results in (c) and (d) in the equation and solve for the magnitude of $E$.

Mayukh Banik

Numerade Educator

Problem 51

Interpreting Gauss Gauss' law states
$$
\oint_{A} \vec{E} \cdot d \vec{A}=q_{A} / \varepsilon_{0}
$$
where $A$ is a surface and $q_{A}$ is a charge.
(a) Which of the following statements are true about the surface $A$ appearing in Gauss' law for the equation to hold? You may list any number of these statements including all or none.
i. The surface $A$ must be a closed surface (must cover a volume).
ii. The surface $A$ must contain all the charges in the problem.
iii. The surface $A$ must be a highly symmetrical surface like a sphere or a cylinder.
iv. The surface $A$ must be a conductor.
v. The surface $A$ is purely imaginary.
vi. The normals to the surface $A$ must all be in the same direction as the electric field on the surface.
(b) Which of the following statements are true about the charge $q_{A}$ appearing in Gauss' law? You may list any number of these statements including all or none.
i. The charge $q_{A}$ must be all the charge lying on the Gaussian surface.
ii. The charge $q_{A}$ must be the charge lying within the Gaussian surface.
iii. The charge $q_{A}$ must be all the charge in the problem.
iv. The charge $q_{A}$ flows onto the Gaussian surface once the surface is established.
v. The electric field $E$ in the integral on the left of Gauss' law is due only to the charge $q_{A}$.
vi. The electric field $E$ in the integral on the left on Gauss' law is due to all charges in the problem.

Yuva S

Numerade Educator