A 56 Cm × 39 Cm Rectangle Lies in the Xy-plane You May Want to Review

6.1 Electrical Flux

xx .

A uniform electric field of magnitude $1.1\times {\mathrm{ten}}^{\mathrm{iv}}\text{North/C}$ is perpendicular to a square sheet with sides ii.0 m long. What is the electric flux through the sheet?

21.

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

22 .

Find the electric flux through a rectangular area $3\text{cm}\times 2\text{cm}$ betwixt two parallel plates where there is a abiding electrical field of 30 Northward/C for the post-obit orientations of the expanse: (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 bending tin as well be given as $180\text{°}+30\text{°}.$

23.

The electric flux through a square-shaped surface area of side v cm near a big charged canvas is found to be $3\times {\mathrm{x}}^{\mathrm{-5}}\text{N}·{\text{m}}^{\mathrm{two}}\text{/}\text{C}$ when the area is parallel to the plate. Find the charge density on the canvass.

24 .

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

25.

A square surface of expanse $2{\text{cm}}^2$ is in a space of compatible electric field of magnitude ${10}^{\mathrm{iii}}\text{N/C}$ . The corporeality of flux through it depends on how the square is oriented relative to the management of the electric field. Observe the electric flux through the foursquare, when the normal to information technology makes the following angles with electric field: (a) $30\text{°}$ , (b) $90\text{°}$ , and (c) $0\text{°}$ . Note that these angles can too be given every bit $180\text{°}+\theta$ .

26 .

A vector field is pointed forth the z-axis, $\overrightarrow{\text{v}}=\frac{\alpha }{x^{\mathrm{two}}+y^2}\widehat{\text{z}}.$ (a) Notice 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 $c<y<d$ . (Get out your answer as an integral.)

27.

Consider the compatible electric field $\overrightarrow{\text{East}}=(4.0\widehat{\text{j}}+\mathrm{three.0}\widehat{\text{k}})\times {\mathrm{ten}}^3\text{N/C}\text{.}$ What is its electric flux through a circular surface area of radius 2.0 yard that lies in the xy-plane?

28 .

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

29.

An infinite charged wire with accuse 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?

half dozen.2 Explaining Gauss's Law

30 .

Make up one's mind the electric flux through each airtight surface where the cantankerous-section within the surface is shown below.

31.

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

32 .

A point charge q is located at the centre of a cube whose sides are of length a. If there are no other charges in this organisation, what is the electrical flux through 1 face of the cube?

33.

A signal accuse of $\mathrm{x}\mu \text{C}$ is at an unspecified location inside a cube of side 2 cm. Observe the net electric flux though the surfaces of the cube.

34 .

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

35.

A accuse q is placed at i of the corners of a cube of side a, as shown beneath. Notice the magnitude of the electric flux through the shaded face due to q. Assume $q>0$ .

36 .

The electric flux through a cubical box eight.0 cm on a side is $\mathrm{ane.2}\times {10}^3\text{Due north}·{\text{m}}^2\text{/C}\text{.}$ What is the total charge enclosed by the box?

37.

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

38 .

A cube whose sides are of length d is placed in a uniform electrical field of magnitude $\mathrm{Due\; east}=4.0\times {\mathrm{ten}}^3\text{N/C}$ so that the field is perpendicular to two contrary faces of the cube. What is the net flux through the cube?

39.

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

40 .

A total charge $5.0\times {\mathrm{ten}}^{\mathrm{-half-dozen}}\text{C}$ is distributed uniformly throughout a cubical volume whose edges are eight.0 cm long. (a) What is the charge density in the cube? (b) What is the electrical flux through a cube with 12.0-cm edges that is concentric with the accuse distribution? (c) Practise 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 iii.0 cm that is too concentric with the charge distribution?

6.3 Applying Gauss's Law

41.

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

42 .

Suppose that the charge density of the spherical charge distribution shown in Figure half dozen.23 is $\rho (r)={\rho }_{0}r\text{/}R$ for $r\le R$ and naught for $r>R.$ Obtain expressions for the electric field both within and exterior the distribution.

43.

A very long, sparse wire has a compatible linear accuse density of $\mathrm{fifty}\mu \text{C/m}\text{.}$ What is the electrical field at a distance two.0 cm from the wire?

44 .

A charge of $\mathrm{-xxx}\mu \text{C}$ 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) v.0 cm, and (c) 20.0 cm from the center of the sphere.

45.

Repeat your calculations for the preceding problem, given that the charge is distributed uniformly over the surface of a spherical conductor of radius ten.0 cm.

46 .

A total charge Q is distributed uniformly throughout a spherical trounce of inner and outer radii $r_{\mathrm{one}}\text{and}{r}_2,$ respectively. Evidence that the electrical field due to the charge is

$\begin{array}{cccc}\overrightarrow{\text{E}}=\overrightarrow{0}\hfill & & & (r\le r_1);\hfill \\ \overrightarrow{\text{E}}=\frac{Q}{\mathrm{four}\pi {\epsilon }_0{r}^{\mathrm{two}}}\left(\frac{r^{\mathrm{iii}}-r_{\mathrm{one}}{}^3}{r_2{}^3-r_{\mathrm{one}}{}^3}\right)\widehat{\text{r}}\hfill & & & (r_1\le r\le r_2);\hfill \\ \overrightarrow{\text{E}}=\frac{Q}{\mathrm{iv}\pi {\epsilon }_0{r}^{\mathrm{two}}}\widehat{\text{r}}\hfill & & & (r\ge r_2).\hfill \end{array}$

47.

When a charge is placed on a metal sphere, it ends up in equilibrium at the outer surface. Use this data to determine the electric field of $+3.0\mu \text{C}$ charge put on a v.0-cm aluminum spherical brawl at the following two points in space: (a) a point i.0 cm from the centre of the ball (an within bespeak) and (b) a point x cm from the eye of the ball (an outside point).

48 .

A large sheet of accuse has a compatible charge density of $10\mu {\text{C/m}}^{\mathrm{ii}}\text{.}$ What is the electric field due to this accuse at a point but above the surface of the canvass?

49.

Determine if gauge cylindrical symmetry holds for the post-obit situations. Country why or why not. (a) A 300-cm long copper rod of radius one cm is charged with +500 nC of charge and we seek electrical field at a signal 5 cm from the center of the rod. (b) A x-cm long copper rod of radius one cm is charged with +500 nC of charge and we seek electrical field at a indicate 5 cm from the centre of the rod. (c) A 150-cm wooden rod is glued to a 150-cm plastic rod to brand a 300-cm long rod, which is then painted with a charged paint so that one obtains a compatible charge density. The radius of each rod is 1 cm, and we seek an electric field at a point that is four cm from the center of the rod. (d) Aforementioned rod as (c), only nosotros seek electric field at a indicate that is 500 cm from the center of the rod.

l .

A long silver rod of radius iii cm has a accuse of $\text{−}\mathrm{v}\mu \text{C/cm}$ 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 electrical field at a point ii cm from the centre of the rod (an inside point).

51.

The electric field at 2 cm from the center of long copper rod of radius one cm has a magnitude 3 Due north/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 reverse sides of the cube perpendicularly?

52 .

A long copper cylindrical vanquish of inner radius ii 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 crush has exactly opposite charge to that of the aluminum rod while the outer surface of the copper shell has the same accuse as the aluminum rod. Find the magnitude and direction of the electric field at points that are at the post-obit distances from the center of the aluminum rod: (a) 0.five cm, (b) 1.5 cm, (c) 2.v cm, (d) iii.5 cm, and (due east) 7 cm.

53.

Accuse is distributed uniformly with a density $\rho$ throughout an infinitely long cylindrical book of radius R. Evidence that the field of this charge distribution is directed radially with respect to the cylinder and that

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

54 .

Charge is distributed throughout a very long cylindrical volume of radius R such that the charge density increases with the altitude r from the central centrality of the cylinder co-ordinate 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

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

55.

The electrical field 10.0 cm from the surface of a copper brawl of radius five.0 cm is directed toward the ball's centre and has magnitude $4.0\times {10}^2\text{N/C}\text{.}$ How much accuse is on the surface of the ball?

56 .

Charge is distributed throughout a spherical shell of inner radius $r_1$ and outer radius $r_2$ with a book density given past $\rho ={\rho }_0{r}_1\text{/}r,$ where ${\rho }_0$ is a constant. Decide the electric field due to this charge every bit a function of r, the distance from the heart of the shell.

57.

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 within and outside the sphere.

58 .

Consider a uranium nucleus to exist sphere of radius $R=7.4\times {10}^{\mathrm{-xv}}\text{grand}$ with a charge of 92due east distributed uniformly throughout its volume. (a) What is the electric force exerted on an electron when it is $3.0\times {10}^{\mathrm{-fifteen}}\text{yard}$ from the centre of the nucleus? (b) What is the acceleration of the electron at this signal?

59.

The book 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 electrical field produced by this charge distribution?

half dozen.4 Conductors in Electrostatic Equilibrium

sixty .

An uncharged conductor with an internal cavity is shown in the following effigy. Apply the airtight surface S along with Gauss' law to bear witness 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 accuse on the outer surface of the conductor?

Figure 6.46 A charge inside a cavity of a metallic. Charges at the outer surface do non depend on how the charges are distributed at the inner surface since Due east field inside the body of the metal is zero.

61.

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 ii cavities by using non-conducting supports. In add-on, a bespeak charge $+q_0$ is placed outside at a altitude r from the center of the sphere. (a) Draw approximate accuse distributions in the metal although metallic sphere has no net charge. (b) Draw electric field lines. Draw enough lines to represent all distinctly different places.

62 .

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

63.

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

64 .

An aluminum spherical brawl of radius 4 cm is charged with $\mathrm{v}\mu \text{C}$ of accuse. A copper spherical beat out of inner radius six cm and outer radius 8 cm surrounds it. A total accuse of $\mathrm{-8}\mu \text{C}$ is put on the copper trounce. (a) Detect the electrical field at all points in space, including points inside the aluminum and copper shell when copper beat out and aluminum sphere are concentric. (b) Find the electric field at all points in infinite, including points inside the aluminum and copper vanquish when the centers of copper shell and aluminum sphere are 1 cm apart.

65.

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

66 .

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

67.

Two parallel plates ten cm on a side are given equal and contrary charges of magnitude $\mathrm{five.0}\times {10}^{\mathrm{-9}}\text{C}\text{.}$ The plates are 1.5 mm apart. What is the electric field at the centre of the region betwixt the plates?

68 .

Ii parallel conducting plates, each of cross-sectional area ${400\text{cm}}^2$ , are 2.0 cm apart and uncharged. If $\mathrm{ane.0}\times {10}^{12}$ electrons are transferred from 1 plate to the other, what are (a) the charge density on each plate? (b) The electric field betwixt the plates?

69.

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

70 .

A signal charge $q=\mathrm{-5.0}\times {10}^{\mathrm{-12}}\text{C}$ is placed at the centre of a spherical conducting shell of inner radius iii.five 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 crush? (b) What is the charge density on the outer surface of the trounce? (c) What is the internet accuse on the usher?

71.

A solid cylindrical usher of radius a is surrounded past a concentric cylindrical vanquish of inner radius b. The solid cylinder and the shell carry charges +Q and –Q, respectively. Assuming that the length Fifty of both conductors is much greater than a or b, decide the electric field equally a function of r, the altitude from the common fundamental centrality of the cylinders, for (a) $r<a;$ (b) $a<r<b;$ and (c) $r>b.$

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Source: https://openstax.org/books/university-physics-volume-2/pages/6-problems

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