A point charge is at the center of an imaginary sphere

Figure 22-37 shows an L-shaped object that has sides which are equal in length. Positive charge is distributed uniformly along the length of the object. What is the direction of the electric field along the dashed line? Explain your answer. SSM

Positive charge is distributed uniformly along the entire length of the axis, and negative charge is distributed uniformly along the entire length of the axis. The charge per unit length on the two axes is identical, except for the sign. Determine the direction of the electric field at points on the lines defined by y _ x and y _ _x. Explain your answer.

True or false: (a) The electric field due to a hollow uniformly charged thin spherical shell is zero at all points inside the shell. (b) In electrostatic equilibrium, the electric field everywhere inside the material of a conductor must be zero. (c) If the net charge on a conductor is zero, the charge density must be zero at every point on the surface of the conductor.

If the electric flux through a closed surface is zero, must the electric field be zero everywhere on that surface? If not, give a specific example. From the given information can the net charge inside the surface be determined? If so, what is it?

True or false: (a) Gausss law holds only for symmetric charge distributions. (b) The result that everywhere inside the material of a conductor under electrostatic conditions can be derived from Gausss law.

A single point charge is located at the center of both an imaginary cube and an imaginary sphere. How does the electric flux through the surface of the cube compare to that through the surface of the sphere? Explain your answer.

An electric dipole is completely inside a closed imaginary surface and there are no other charges. True or false: (a) The electric field is zero everywhere on the surface. (b) The electric field is normal to the surface everywhere on the surface. (c) The electric flux through the surface is zero. (d) The electric flux through the surface could be positive or negative. (e) The electric flux through a portion of the surface might not be zero.

Explain why the electric field strength increases linearly with , rather than decreases inversely with , between the center and the surface of a uniformly charged solid sphere.

Suppose that the total charge on the conducting spherical shell in Figure 22-38 is zero. The negative point charge at the center has a magnitude given by What is the direction of the electric field in the following regions? (a) (b) (c) and Explain your answer.

The conducting shell in Figure 22-38 is grounded, and the negative point charge at the center has a magnitude given by Which of the following statements is correct? (a) The charge on the inner surface of the shell is and the charge on the outer surface is (b) The charge on the inner surface of the shell is and the charge on the outer surface is zero. (c) The charge on both surfaces of the shell is (d) The charge on both surfaces of the shell is zero. _Q. _Q _Q. _Q Q. r R SSM 2 R . 2 r R1 , r R1, Q.

The conducting shell in Figure 22-38 is grounded, and the negative point charge at the center has a magnitude given by What is the direction of the electric field in the following regions? (a) (b) (c) and Explain your answers.

In the chapter, the expression for the electric field due to a uniformly charged disk (on its axis), was derived. At any location on the axis, the field magnitude is given by At large distances it was shown that this equation approaches Very near the disk the field strength is approximately that of an infinite plane of charge or Suppose you have a disk of radius 2.5 cm that has a uniform surface charge density of Use both the exact and approximate expression from those given above to find the electric field strength on the axis at distances of (a) 0.010 cm, (b) 0.040 cm, and (c) 5.0 m. Compare the two values in each case and comment on how well the approximations work in their region of validity.

A uniform line charge that has a linear charge density equal to 3.5 nC>m is on the x axis between x _ 0 and x _ 5.0 m. (a) What is its total charge? Find the electric field on the axis at (b) (c) and (d) (e) Estimate the electric field at using the approximation that the charge is a point charge on the axis at and compare your result with the result calculated in Part (d). (To do this, you will need to assume that the values given in this problem statement are valid to more than two significant figures.) Is your approximate result greater or smaller than the exact result? Explain your answer.

Two infinite nonconducting sheets of charge are parallel to each other, with sheet A in the plane and sheet B in the plane. Find the electric field in the region in the region and between the sheets for the following situations. (a) When each sheet has a uniform surface charge density equal to and (b) when sheet Ahas a uniform surface charge density equal to and sheet B has a uniform surface charge density equal to (c) Sketch the electric field line pattern for each case.

A charge of is uniformly distributed on a ring of radius 8.5 cm. Find the electric field strength on the axis at distances of (a) 1.2 cm, (b) 3.6 cm, and (c) 4.0 m from the center of the ring. (d) Find the field strength at 4.0 m using the approximation that the ring is a point charge at the origin, and compare your results for Parts (c) and (d). Is your approximate result a good one? Explain your answer.

A nonconducting disk of radius lies in the plane with its center at the origin. The disk has a uniform surface charge density Find the value of for which Note that at this distance, the magnitude of the electric field strength is half the electric field strength at points on the axis that are very close to the disk.

Aring that has radius a lies in the plane with its center at the origin. The ring is uniformly charged and has a total charge Find on the axis at (a) (b) (c) (d) and (e) (f) Use your results to plot versus for both positive and negative values of (Assume that these distances are exact.)

A nonconducting disk of radius a lies in the plane with its center at the origin. The disk is uniformly charged and has a total charge Find on the axis at (a) (b) (c) (d) and (e) (f) Use your results to plot versus for both positive and negative values of (Assume that these distances are exact.)

SPREADSHEET (a) Using a spreadsheet program or graphing calculator, make a graph of the electric field strength on the axis of a disk that has a radius and a surface charge density (b) Compare your results to the results based on the approximation (the formula for the electric field strength of a uniformly charged infinite sheet). At what distance does the solution based on approximation differ from the exact solution by 10.0 percent?

(a) Show that the electric field strength on the axis of a ring charge of radius a has maximum values at (b) Sketch the field strength versus for both positive and negative values of (c) Determine the maximum value of

A line charge that has a uniform linear charge density lies along the axis from to where Show that the component of the electric field at a point on the axis is given by where and y _ 0.

A ring of radius a has a charge distribution on it that varies as as shown in Figure 22-39. (a) What is the direction of the electric field at the center of the ring? (b) What is the magnitude of the field at the center of the ring?

A line of charge that has uniform linear charge density lies on the axis from to Show that the component of the electric field at a point on the axis is given by

Calculate the electric field a distance from a uniformly charged infinite flat nonconducting sheet by modeling the sheet as a continuum of infinite straight lines of charge.

Calculate the electric field a distance from a uniformly charged infinite flat nonconducting sheet by modeling the sheet as a continuum of infinite circular rings of charge.

Athin hemispherical shell of radius has a uniform surface charge Find the electric field at the center of the base of the hemispherical shell.

A square that has 10-cm-long edges is centered on the axis in a region where there exists a uniform electric field given by (a) What is the electric flux of this electric field through the surface of a square if the normal to the surface is in the + direction? (b) What is the electric flux through the same square surface if the normal to the surface makes a angle with the axis and an angle of with the axis?

A single point charge is fixed at the origin. An imaginary spherical surface of radius 3.00 m is centered on the axis at (a) Sketch electric field lines for this charge (in two dimensions) assuming twelve equally spaced field lines in the plane leave the charge location, with one of the lines in the direction. Do any lines enter the spherical surface? If so, how many? (b) Do any lines leave the spherical surface? If so, how many? (c) Counting the lines that enter as negative and the ones that leave as positive, what is the net number of field lines that penetrate the spherical surface? (d) What is the net electric flux through this spherical surface?

An electric field is given by where sign equals if if and if Acylinder of length 20 cm and radius 4.0 cm has its center at the origin and its axis along the x axis such that one end is at and the other is at (a) What is the electric flux through each end? (b) What is the electric flux through the curved surface of the cylinder? (c) What is the electric flux through the entire closed surface? (d) What is the net charge inside the cylinder?

Careful measurement of the electric field at the surface of a black box indicates that the net outward electric flux through the surface of the box is (a) What is the net charge inside the box? (b) If the net outward electric flux through the surface of the box were zero, could you conclude that there were no charges inside the box? Explain your answer.

A point charge is at the center of an imaginary sphere that has a radius equal to 0.500 m. (a) Find the surface area of the sphere. (b) Find the magnitude of the electric field at all points on the surface of the sphere. (c) What is the flux of the electric field through the surface of the sphere? (d) Would your answer to Part (c) change if the point charge were moved so that it was inside the sphere but not at its center? (e) What is the flux of the electric field through the surface of an imaginary cube that has 1.00-m-long edges and encloses the sphere?

What is the electric flux through one side of a cube that has a single point charge of placed at its center? Hint: You do not need to integrate any equations to get the answer.

A single point charge is placed at the center of an imaginary cube that has 20-cm-long edges. The electric flux out of one of the cubes sides is How much charge is at the center?

Because the formulas for Newtons law of gravity and for Coulombs law have the same inverse-square dependence on distance, a formula analogous to the formula for Gausss law can be found for gravity. The gravitational field at a location is the force per unit mass on a test mass placed at that location. (Then, for a point mass at the origin, the gravitational field at some position is Compute the flux of the gravitational field through a spherical surface of radius centered at the origin, and verify that the gravitational analog of Gausss law is

An imaginary right circular cone (Figure 22-40) that has a base angle and a base radius is in charge free region that has a uniform electric field (field lines are vertical and parallel to the cones axis). What is the ratio of the number of field lines per unit area penetrating the base to the number of field lines per unit area penetrating the conical surface of the cone? Use Gausss law in your answer. (The field lines in the figure are only a representative sample.)

In the atmosphere and at an altitude of 250 m, you measure the electric field to be directed downward, and you measure the electric field to be directed downward at an altitude of 400 m. Calculate the volume charge density of the atmosphere in the region between altitudes of 250 m and 400 m, assuming it to be uniform. (You may neglet the curvature of Earth. Why?)

A thin nonconducting spherical shell of radius has a total charge that is uniformly distributed on its surface. Asecond, larger thin nonconducting spherical shell of radius that is coaxial with the first has a charge that is uniformly distributed on its surface. (a) Use Gausss law to obtain expressions for the electric field in each of the three regions: and (b) What should the ratio of the charges and the relative signs for and be for the electric field to be zero throughout the region (c) Sketch the electric field lines for the situation in Part (b) when is positive.

A nonconducting thin spherical shell of radius 6.00 cm has a uniform surface charge density of (a) What is the total charge on the shell? Find the electric field at the following distances from the spheres center: (b) 2.00 cm, (c) 5.90 cm, (d) 6.10 cm, and (e) 10.0 cm.

A nonconducting sphere of radius 6.00 cm has a uniform volume charge density of (a) What is the total charge on the sphere? Find the electric field at the following distances from the spheres center: (b) 2.00 cm, (c) 5.90 cm, (d) 6.10 cm, and (e) 10.0 cm.

Consider the solid conducting sphere and the concentric conducting spherical shell in Figure 22-41. The spherical shell has a charge The solid sphere has a charge . (a) How much charge is on the outer surface and how much charge is on the inner surface of the spherical shell? (b) Suppose a metal wire is now connected between the solid sphere and the shell. After electrostatic equilibrium is reestablished, how much charge is on the solid sphere and on each surface of the spherical shell? Does the electric field at the surface of the solid sphere change when the wire is connected? If so, in what way? (c) Suppose we return to the conditions in Part (a) , with on the solid sphere and on the spherical shell. We next connect the solid sphere to ground with a metal wire, and then disconnect it. Then how much total charge is on the solid sphere and on each surface of the spherical shell?

A nonconducting solid sphere of radius 10.0 cm has a uniform volume charge density. The magnitude of the electric field at 20.0 cm from the spheres center is (a) What is the spheres volume charge density? (b) Find the magnitude of the electric field at a distance of 5.00 cm from the spheres center.

A nonconducting solid sphere of radius has a volume charge density that is proportional to the distance from the center. That is, for where is a constant. (a) Find the total charge on the sphere. (b) Find the expressions for the electric field inside the sphere and outside the sphere . (c) Sketch the magnitude of the electric field as a function of the distance from the spheres center.

A sphere of radius has volume charge density for where is a constant and for (a) Find the total charge on the sphere. (b) Find the expressions for the electric field inside and outside the charge distribution. (c) Sketch the magnitude of the electric field as a function of the distance from the spheres center.

A sphere of radius has volume charge density for where is a constant and for (a) Find the total charge on the sphere. (b) Find the expressions for the electric field inside and outside the charge distribution. (c) Sketch the magnitude of the electric field as a function of the distance from the spheres center.

A nonconducting spherical shell of inner radius and outer radius has a uniform volume charge density (a) Find the total charge on the shell. (b) Find expressions for the electric field everywhere.

CONTEXT-RICH, ENGINEERING APPLICATION For your senior project, you are designing a Geiger tube for detecting radiation in the nuclear physics laboratory. This instrument will consist of a long metal cylindrical tube that has a long straight metal wire running down its central axis. The diameter of the wire will be 0.500 mm and the inside diameter of the tube will be 4.00 cm. The tube is to be filled with a dilute gas in which an electrical discharge (breakdown of the gas) occurs when the electric field reaches Determine the maximum linear charge density on the wire if breakdown of the gas is not to happen. Assume that the tube and the wire are infinitely long.

In Problem 46, suppose ionizing radiation produces an ion and an electron at a distance of 1.50 cm from the long axis of the central wire of the Geiger tube. Suppose that the central wire is positively charged and has a linear charge density equal to (a) In this case, what will be the electrons speed as it impacts the wire? (b) How will the electrons speed compare to the ions final speed when it impacts the outside cylinder? Explain your answer.

Show that the electric field due to an infinitely long, uniformly charged thin cylindrical shell of radius a having a surface charge density is given by the following expressions: for and for

A thin cylindrical shell of length 200 m and radius 6.00 cm has a uniform surface charge density of (a) What is the total charge on the shell? Find the electric field at the following radial distances from the long axis of the cylinder: (b) 2.00 cm, (c) 5.90 cm, (d) 6.10 cm, and (e) 10.0 cm. (Use the results of Problem 48.)

An infinitely long nonconducting solid cylinder of radius a has a uniform volume charge density of Show that the electric field is given by the following expressions: for and for where is the distance from the long axis of the cylinder.

A solid cylinder of length \(200 \mathrm{~m}\) and radius \(6.00 \mathrm{~cm}\) has a uniform volume charge density of \(300 \mathrm{nC} / \mathrm{m}^3\). (a) What is the total charge of the cylinder? Use the formulas given in Problem 50 to calculate the electric field at a point equidistant from the ends at the following radial distances from the cylindrical axis: (b) \(2.00 \mathrm{~cm}\), (c) \(5.90 \mathrm{~cm}\), (d) \(6.10 \mathrm{~cm}\), and (e) \(10.0 \mathrm{~cm}\).

Consider two infinitely long, coaxial thin cylindrical shells. The inner shell has a radius \(a_1\) and has a uniform surface charge density of \(\sigma_1\), and the outer shell has a radius \(a_2\) and has a uniform surface charge density of \(\sigma_2\). (a) Use Gauss’s law to find expressions for the electric field in the three regions: \(0 \leq R <a_1 , a_1 <R < a_2\), and \(R > a_2\) where R is the distance from the axis. (b) What is the ratio of the surface charge densities \(\sigma_2 / \sigma_1\) and their relative signs if the electric field is to be zero everywhere outside the largest cylinder? (c) For the case in Part (b), what would be the electric field between the shells? (d) Sketch the electric field lines for the situation in Part (b) if \(\sigma_1\) is positive.

Figure 22-42 shows a portion of an infinitely long, concentric cable in cross section. The inner conductor has a linear charge density of and the outer conductor has no net charge. (a) Find the electric field for all values of where is the perpendicular distance from the common axis of the cylindrical system. (b) What are the surface charge densities on the inside and the outside surfaces of the outer conductor?

An infinitely long nonconducting solid cylinder of radius a has a nonuniform volume charge density. This density varies linearly with the perpendicular distance from its axis, according to where is a constant. (a) Show that the linear charge density of the cylinder is given by (b) Find expressions for the electric field for and

An infinitely long nonconducting solid cylinder of radius a has a nonuniform volume charge density. This density varies with the perpendicular distance from its axis, according to where is a constant. (a) Show that the linear charge density of the cylinder is given by (b) Find expressions for the electric field for and

An infinitely long, nonconducting cylindrical shell of inner radius and outer radius has a uniform volume charge density Find expressions for the electric field everywhere.

The inner cylinder of Figure 22-42 is made of nonconducting material and has a volume charge distribution given by where The outer cylinder is metallic, and both cylinders are infinitely long. (a) Find the charge per unit length (that is, the linear charge density) on the inner cylinder. (b) Calculate the electric field for all values of ELECTRIC CHARGE AND FIELD AT CONDUCTOR SURFACES

An uncharged penny is in a region that has a uniform electric field of magnitude directed perpendicular to its faces. (a) Find the charge density on each face of the penny, assuming the faces are planes. (b) If the radius of the penny is 1.00 cm, find the total charge on one face.

A thin metal slab has a net charge of zero and has square faces that have 12-cm-long sides. It is in a region that has a uniform electric field that is perpendicular to its faces. The total charge induced on one of the faces is 1.2 nC. What is the magnitude of the electric field?

A charge of is uniformly distributed on a thin square sheet of nonconducting material of edge length 20.0 cm. (a) What is the surface charge density of the sheet? (b) What are the magnitude and direction of the electric field next to the sheet and proximate to the center of the sheet?

A conducting spherical shell that has zero net charge has an inner radius and an outer radius A positive point charge is placed at the center of the shell. (a) Use Gausss law and the properties of conductors in electrostatic equilibrium to find the electric field in the three regions: and where is the distance from the center. (b) Draw the electric field lines in all three regions. (c) Find the charge density on the inner surface and on the outer surface of the shell.

The electric field just above the surface of Earth has been measured to typically be pointing downward. (a) What is the sign of the net charge on Earths surface under typical conditions? (b) What is the total charge on Earths surface implied by this measurement?

Apositive point charge of is at the center of a conducting spherical shell that has a net charge of zero, an inner radius equal to 60 cm, and an outer radius equal to 90 cm. (a) Find the charge densities on the inner and outer surfaces of the shell and the total charge on each surface. (b) Find the electric field everywhere. (c) Repeat Part (a) and Part (b) with a net charge of placed on the shell.

If the magnitude of an electric field in air is as great as the air becomes ionized and begins to conduct electricity. This phenomenon is called dielectric breakdown. A charge of is to be placed on a conducting sphere. What is the minimum radius of a sphere that can hold this charge without breakdown?

A thin square conducting sheet that has 5.00-m-long edges has a net charge of The square is in the plane and is centered at the origin. (Assume the charge on each surface is uniformly distributed.) (a) Find the charge density on each side of the sheet and find the electric field on the axis in the region (b) Athin but infinite nonconducting sheet that has a uniform charge density of is now placed in the plane. Find the electric field on the x axis on each side of the square sheet in the region Find the charge density on each surface of the square sheet.

Consider the concentric metal sphere and spherical shells that are shown in Figure 22-43. The innermost is a solid sphere that has a radius R A spherical shell surrounds the sphere and has an inner radius and an outer radius The sphere and the shell are both surrounded by a second spherical shell that has an inner radius and an outer radius None of the three objects initially have a net charge. Then, a negative charge is placed on the inner sphere and a positive charge is placed on the outermost shell. (a) After the charges have reached equilibrium, what will be the direction of the electric field between the inner sphere and the middle shell? (b) What will be the charge on the inner surface of the middle shell? (c) What will be the charge on the outer surface of the middle shell? (d) What will be the charge on the inner surface of the outermost shell? (e) What will be the charge on the outer surface of the outermost shell? (f) Plot as a function of for all values of

A large, flat, nonconducting, nonuniformly charged surface lies in the plane. At the origin, the surface charge density is Asmall distance away from the surface on the positive axis, the component of the electric field is What is a small distance away from the surface on the negative axis?

An infinitely long line charge that has a uniform linear charge density equal to lies parallel to the axis at Apositive point charge that has a magnitude equal to is located at Find the electric field at

A thin, nonconducting, uniformly charged spherical shell of radius (Figure 22-44a) has a total positive charge of A small circular plug is removed from the surface. (a) What are the magnitude and direction of the electric field at the center of the hole? (b) The plug is now put back in the hole (Figure 22-44b). Using the result of Part (a), find the electric force acting on the plug. (c) Using the magnitude of the force, calculate the electrostatic pressure (force/unit area) that tends to expand the sphere. SSM

An infinite thin sheet in the plane has a uniform surface charge density A second infinite thin sheet has a uniform charge density and intersects the plane at the axis and makes an angle of with the plane, as shown in Figure 22-45. Find the electric field at (a) and

Two identical square parallel metal plates each have an area of They are separated by 1.50 cm. They are both initially uncharged. Now a charge of is transferred from the plate on the left to the plate on the right and the charges then establish electrostatic equilibrium. (Neglect edge effects.) (a) What is the electric field between the plates at a distance of 0.25 cm from the plate on the right? (b) What is the electric field between the plates a distance of 1.00 cm from the plate on the left? (c) What is the electric field just to the left of the plate on the left? (d) What is the electric field just to the right of the plate on the right?

Two infinite nonconducting uniformly charged planes lie parallel to each other and to the plane. One is at and has a surface charge density of The other is at and has a surface charge density of Find the electric field in the region: (a) (b) and (c)

A quantum-mechanical treatment of the hydrogen atom shows that the electron in the atom can be treated as a smeared-out distribution of negative charge of the form Here represents the distance from the center of the nucleus and a represents the first Bohr radius, which has a numerical value of 0.0529 nm. Recall that the nucleus of a hydrogen atom consists of just one proton and treat this proton as a positive point charge. (a) Calculate using the fact that the atom is neutral. (b) Calculate the electric field at any distance from the nucleus.

A uniformly charged ring has a radius , lies in a horizontal plane, and has a negative charge given by A small particle of mass has a positive charge given by The small particle is located on the axis of the ring. (a) What is the minimum value of such that the particle will be in equilibrium under the action of gravity and the electrostatic force? (b) If is twice the value calculated in Part (a) , where will the particle be when it is in equilibrium? Express your answer in terms of

A long, thin, nonconducting plastic rod is bent into a circular loop that has a radius a. Between the ends of the rod a short gap of length where remains. Apositive charge of magnitude is evenly distributed on the loop. (a) What is the direction of the electric field at the center of the loop? Explain your answer. (b) What is the magnitude of the electric field at the center of the loop?

A nonconducting solid sphere that is 1.20 m in diameter and has its center on the axis at has a uniform volume charge of density of Concentric with the sphere is a thin nonconducting spherical shell that has a diameter of 2.40 m and a uniform surface charge density of Calculate the magnitude and direction of the electric field at (a) (b) and (c)

An infinite nonconducting plane sheet of charge that has a surface charge density lies in the plane. A second infinite nonconducting plane sheet of charge that has a surface charge density of lies in the plane. Lastly, a nonconducting thin spherical shell that has a radius of 1.00 m and its center in the plane at the intersection of the two charged planes has a surface charge density of Find the magnitude and direction of the electric field on the axis at (a) and (b)

An infinite nonconducting plane sheet lies in the plane and has a uniform surface charge density of An infinite nonconducting line charge of uniform linear charge density passes through the origin at an angle of with the axis in the plane. A solid nonconducting sphere of volume charge density and radius 0.800 m is centered on the axis at Calculate the magnitude and direction of the electric field in the plane at

A uniformly charged, infinitely long line of negative charge has a linear charge density of and is located on the axis. A small positively charged particle that has a mass and a charge is in a circular orbit of radius in the plane centered on the line of charge. (a) Derive an expression for the speed of the particle. (b) Obtain an expression for the period of the particles orbit. SSM

A stationary ring of radius lies in the plane and has a uniform positive charge A small particle that has mass and a negative charge is located at the center of the ring. (a) Show that if the electric field along the axis of the ring is proportional to (b) Find the force on the particle as a function of (c) Show that if the particle is given a small displacement in the direction, it will perform simple harmonic motion. (d) What is the frequency of that motion?

The charges and of Problem 80 are and respectively, and the radius of the ring is 8.00 cm. When the particle is given a small displacement in the direction, it oscillates about its equilibrium position at a frequency of 3.34 Hz. (a) What is the particles mass? (b) What is the frequency if the radius of the ring is doubled to 16.0 cm and all other parameters remain unchanged?

If the radius of the ring in Problem 80 is doubled while keeping the linear charge density on the ring the same, does the frequency of oscillation of the particle change? If so, by what factor does it change?

A uniformly charged nonconducting solid sphere of radius has its center at the origin and has a volume charge density of (a) Show that at a point within the sphere a distance from the center (b) Material is removed from the sphere leaving a spherical cavity that has a radius and its center at on the axis (Figure 22-46). Calculate the electric field at points 1 and 2 shown in Figure 22-46. Hint: Model the sphere-with-cavity as two uniform spheres of equal positive and negative charge densities.

Show that the electric field throughout the cavity of Problem 83b is uniform and is given by

The cavity in Problem 83b is now filled with a uniformly charged nonconducting material with a total charge of Calculate the new values of the electric field at points 1 and 2 shown in Figure 22-46.

A small Gaussian surface in the shape of a cube has faces parallel to the and planes (Figure 22-47) and is in a region in which the electric field is parallel to the axis. (a) Using the differential approximation, show that the net electric flux of the electric field out of the Gaussian surface is given by , where is the volume enclosed by the Gaussian surface. (b) Using Gausss law and the results of Part (a) show that where is the volume charge density inside the cube. (This equation is the one-dimensional version of the point form of Gausss law.)

Consider a simple but surprisingly accurate model for the hydrogen molecule: two positive point charges, each having charge are placed inside a uniformly charged sphere of radius which has a charge equal to The two point charges are placed symmetrically, equidistant from the center of the sphere (Figure 22-48). Find the distance from the center, where the net force on either point charge is zero. SSM

An electric dipole that has a dipole moment of is located at a perpendicular distance from an infinitely long line charge that has a uniform linear charge density Assume that the dipole moment is in the same direction as the field of the line of charge. Determine an expression for the electric force on the dipole.