Chapter 22

To be able to calculate the electric field created by a known distribution of charge using Gauss's Law, which of the following must be true? a) The charge distribution must be in a nonconducting medium. b) The charge distribution must be in a conducting medium. c) The charge distribution must have spherical or cylindrical symmetry. d) The charge distribution must be uniform. e) The charge distribution must have a high degree of symmetry that allows assumptions about the symmetry of its electric field to be made.

An electric dipole consists of two equal and opposite charges situated a very small distance from each other. When the dipole is placed in a uniform electric field, which of the following statements is true? a) The dipole will not experience any net force from the electric field; since the charges are equal and have opposite signs, the individual effects will cancel out. b) There will be no net force and no net torque acting on the dipole. c) There will be a net force but no net torque acting on the dipole. d) There will be no net force, but there will (in general) be a net torque acting on dipole.

A point charge, \(+Q\), is located on the \(x\) -axis at \(x=a\), and a second point charge, \(-Q\), is located on the \(x\) -axis at \(x=-a\). A Gaussian surface with radius \(r=2 a\) is centered at the origin. The flux through this Gaussian surface is a) zero. b) greater than zero. c) less than zero. d) none of the above.

A charge of \(+2 q\) is placed at the center of an uncharged conducting shell. What will be the charges on the inner and outer surfaces of the shell, respectively? a) \(-2 q,+2 q\) b) \(-q,+q\) c) \(-2 q,-2 q\) d) \(-2 q,+4 q\)

At which of the following locations is the electric field the strongest? a) a point \(1 \mathrm{~m}\) from a \(1 \mathrm{C}\) point charge b) a point \(1 \mathrm{~m}\) (perpendicular distance) from the center of a \(1-\mathrm{m}\) -long wire with \(1 \mathrm{C}\) of charge distributed on it c) a point \(1 \mathrm{~m}\) (perpendicular distance) from the center of a \(1-\mathrm{m}^{2}\) sheet of charge with \(1 \mathrm{C}\) of charge distributed on it d) a point \(1 \mathrm{~m}\) from the surface of a charged spherical shell of charge \(1 \mathrm{C}\) with a radius of \(1 \mathrm{~m}\) e) a point \(1 \mathrm{~m}\) from the surface of a charged spherical shell of charge \(1 \mathrm{C}\) with a radius of \(0.5 \mathrm{~m}\)

The electric flux through a spherical Gaussian surface of radius \(R\) centered on a charge \(Q\) is \(1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right) .\) What is the electric flux through a cubic Gaussian surface of side \(R\) centered on the same charge \(Q ?\) a) less than \(1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right)\) b) more than \(1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right)\) c) equal to \(1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right)\) d) cannot be determined from the information given

A single positive point charge, \(q,\) is at one corner of a cube with sides of length \(L\), as shown in the figure. The net electric flux through the three net electric flux through the three adjacent sides is zero. The net electric flux through each of the other three sides is a) \(q / 3 \epsilon_{0}\). b) \(q / 6 \epsilon_{0}\). c) \(q / 24 \epsilon_{0}\). d) \(q / 8 \epsilon_{0}\).

Three \(-9-\mathrm{mC}\) point charges are located at (0,0) \((3 \mathrm{~m}, 3 \mathrm{~m})\), and \((3 \mathrm{~m},-3 \mathrm{~m})\). What is the magnitude of the electric field at \((3 \mathrm{~m}, 0) ?\) a) \(0.9 \cdot 10^{7} \mathrm{~N} / \mathrm{C}\) b) \(1.2 \cdot 10^{7} \mathrm{~N} / \mathrm{C}\) c) \(1.8 \cdot 10^{7} \mathrm{~N} / \mathrm{C}\) d) \(2.4 \cdot 10^{7} \mathrm{~N} / \mathrm{C}\) e) \(3.6 \cdot 10^{7} \mathrm{~N} / \mathrm{C}\) f) \(5.4 \cdot 10^{7} \mathrm{~N} / \mathrm{C}\) g) \(10.8 \cdot 10^{7} \mathrm{~N} / \mathrm{C}\)

Which of the following statements is (are) true? a) There will be no change in the charge on the inner surface of a hollow conducting sphere if additional charge is placed on the outer surface. b) There will be some change in the charge on the inner surface of a hollow conducting sphere if additional charge is placed on the outer surface. c) There will be no change in the charge on the inner surface of a hollow conducting sphere if additional charge is placed at the center of the sphere. d) There will be some change in the charge on the inner surface of a hollow conducting sphere if additional charge is placed at the center of the sphere.

Many people had been sitting in a car when it was struck by lightning. Why were they able to survive such an experience?

Why is it a bad idea to stand under a tree in a thunderstorm? What should one do instead to avoid getting struck by lightning?

Why do electric field lines never cross?

How is it possible that the flux through a closed surface does not depend on where inside the surface the charge is located (that is, the charge can be moved around inside the surface with no effect whatsoever on the flux)? If the charge is moved from just inside to just outside the surface, the flux changes discontinuously to zero, according to Gauss's Law. Does this really happen? Explain.

A solid conducting sphere of radius \(r_{1}\) has a total charge of \(+3 Q .\) It is placed inside (and concentric with) a conducting spherical shell of inner radius \(r_{2}\) and outer radius \(r_{3}\). Find the electric field in these regions: \(rr_{3}\).

A dipole is completely enclosed by a spherical surface. Describe how the total electric flux through this surface varies with the strength of the dipole.

Repeat Example 22.3 , assuming that the charge distribution is \(-\lambda\) for \(-a

Saint Elmos fire is an eerie glow that appears at the tips of masts and yardarms of sailing ships in stormy weather and at the tips and edges of the wings of aircraft in flight. St. Elmo's fire is an electrical phenomenon. Explain it, concisely.

A point charge, \(q=4.00 \cdot 10^{-9} \mathrm{C},\) is placed on the \(x\) -axis at the origin. What is the electric field produced at \(x=25.0 \mathrm{~cm} ?\)

\( \mathrm{~A}+48.00-\mathrm{nC}\) point charge is placed on the \(x\) -axis at \(x=4.000 \mathrm{~m},\) and \(\mathrm{a}-24.00-\mathrm{n} \mathrm{C}\) point charge is placed on the \(y\) -axis at \(y=-6.000 \mathrm{~m} .\) What is the direction of the electric field at the origin?

Three charges are on the \(y\) -axis. Two of the charges, each \(-q,\) are located \(y=\pm d,\) and the third charge, \(+2 q,\) is located at \(y=0 .\) Derive an expression for the electric field at a point \(P\) on the \(x\) -axis.

Consider an electric dipole on the \(x\) -axis and centered at the origin. At a distance \(h\) along the positive \(x\) -axis, the magnitude of electric field due to the electric dipole is given by \(k(2 q d) / h^{3} .\) Find a distance perpendicular to the \(x\) axis and measured from the origin at which the magnitude of the electric field stays the same.

A charge per unit length \(+\lambda\) is uniformly distributed along the positive \(y\) -axis from \(y=0\) to \(y=+a\). A charge per unit length \(-\lambda\) is uniformly distributed along the negative \(y\) axis from \(y=0\) to \(y=-a\). Write an expression for the electric field (magnitude and direction) at a point on the \(x\) -axis a distance \(x\) from the origin.

A thin glass rod is bent into a semicircle of radius \(R\). A charge \(+Q\) is uniformly distributed along the upper half, and a charge \(-Q\) is uniformly distributed along the lower half as shown in the figure. Find the magnitude and direction of the electric field \(\vec{E}\) (in component form) at point \(P\), the center of the semicircle.

Two uniformly charged insulating rods are bent in a semicircular shape with radius \(r=10.0 \mathrm{~cm} .\) If they are positioned so they form a circle but do not touch and have opposite charges of \(+1.00 \mu \mathrm{C}\) and \(-1.00 \mu \mathrm{C}\) find the magnitude and direction of the electric field at the center of the composite circular charge configuration.

A uniformly charged rod of length \(L\) with total charge \(Q\) lies along the \(y\) -axis, from \(y=0\) to \(y=L\). Find an expression for the electric field at the point \((d, 0)\) (that is, the point at \(x=d\) on the \(x\) -axis).

A thin, flat washer is a disk with an outer diameter of \(10.0 \mathrm{~cm}\) and a hole in the center with a diameter of \(4.00 \mathrm{~cm} .\) The washer has a uniform charge distribution and a total charge of \(7.00 \mathrm{nC}\). What is the electric field on the axis of the washer at a distance of \(30.0 \mathrm{~cm}\) from the center of the washer?

Research suggests that the electric fields in some thunderstorm clouds can be on the order of \(10.0 \mathrm{kN} / \mathrm{C}\). Calculate the magnitude of the electric force acting on a particle with two excess electrons in the presence of a \(10.0-\mathrm{kN} / \mathrm{C}\) field.

Electric dipole moments of molecules are often measured in debyes \((\mathrm{D}),\) where \(1 \mathrm{D}=3.34 \cdot 10^{-30} \mathrm{C} \mathrm{m} .\) For instance, the dipole moment of hydrogen chloride gas molecules is \(1.05 \mathrm{D}\). Calculate the maximum torque such a molecule can experience in the presence of an electric field of magnitude \(160.0 \mathrm{~N} / \mathrm{C}\).

An electron is observed traveling at a speed of \(27.5 \cdot 10^{6} \mathrm{~m} / \mathrm{s}\) parallel to an electric field of magnitude \(11,400 \mathrm{~N} / \mathrm{C}\) How far will the electron travel before coming to a stop?

Two charges, \(+e\) and \(-e,\) are a distance of \(0.68 \mathrm{nm}\) apart in an electric field, \(E,\) that has a magnitude of \(4.4 \mathrm{kN} / \mathrm{C}\) and is directed at an angle of \(45^{\circ}\) with the dipole axis. Calculate the dipole moment and thus the torque on the dipole in the electric field.

A body of mass \(M\), carrying charge \(Q\), falls from rest from a height \(h\) (above the ground) near the surface of the Earth, where the gravitational acceleration is \(g\) and there is an electric field with a constant component \(E\) in the vertical direction. a) Find an expression for the speed, \(v,\) of the body when it reaches the ground, in terms of \(M, Q, h, g,\) and \(E\). b) The expression from part (a) is not meaningful for certain values of \(M, g, Q,\) and \(E\). Explain what happens in such cases.

A water molecule, which is electrically neutral but has a dipole moment of magnitude \(p=6.20 \cdot 10^{-30} \mathrm{C} \mathrm{m},\) is \(1.00 \mathrm{~cm}\) away from a point charge \(q=+1.00 \mu \mathrm{C} .\) The dipole will align with the electric field due to the charge. It will also experience a net force, since the field is not uniform. a) Calculate the magnitude of the net force. (Hint: You do not need to know the precise size of the molecule, only that it is much smaller than \(1 \mathrm{~cm} .)\) b) Is the molecule attracted to or repelled by the point charge? Explain.

A total of \(3.05 \cdot 10^{6}\) electrons are placed on an initially uncharged wire of length \(1.33 \mathrm{~m}\). a) What is the magnitude of the electric field a perpendicular distance of \(0.401 \mathrm{~m}\) away from the midpoint of the wire? b) What is the magnitude of the acceleration of a proton placed at that point in space? c) In which direction does the electric field force point in this case?

Four charges are placed in a three-dimensional space. The charges have magnitudes \(+3 q,-q,+2 q,\) and \(-7 q .\) If a Gaussian surface encloses all the charges, what will be the electric flux through that surface?

Consider a hollow spherical conductor with total charge \(+5 e\). The outer and inner radii are \(a\) and \(b\), respectively. (a) Calculate the charge on the sphere's inner and outer surfaces if a charge of \(-3 e\) is placed at the center of the sphere. (b) What is the total net charge of the sphere?

A spherical aluminized Mylar balloon carries a charge \(Q\) on its surface. You are measuring the electric field at a distance \(R\) from the balloon's center. The balloon is slowly inflated, and its radius approaches but never reaches R. What happens to the electric field you measure as the balloon increases in radius. Explain.

A hollow conducting spherical shell has an inner radius of \(8.00 \mathrm{~cm}\) and an outer radius of \(10.0 \mathrm{~cm} .\) The electric field at the inner surface of the shell, \(E_{\mathrm{i}}\), has a magnitude of \(80.0 \mathrm{~N} / \mathrm{C}\) and points toward the center of the sphere, and the electric field at the outer surface, \(E_{\infty}\) has a magnitude of \(80.0 \mathrm{~N} / \mathrm{C}\) and points away from the center of the sphere (see the figure). Determine the magnitude of the charge on the inner surface and the outer surface of the spherical shell.

A \(-6.00-n C\) point charge is located at the center of a conducting spherical shell. The shell has an inner radius of \(2.00 \mathrm{~m},\) an outer radius of \(4.00 \mathrm{~m},\) and a charge of \(+7.00 \mathrm{nC}\) a) What is the electric field at \(r=1.00 \mathrm{~m} ?\) b) What is the electric field at \(r=3.00 \mathrm{~m} ?\) c) What is the electric field at \(r=5.00 \mathrm{~m} ?\) d) What is the surface charge distribution, \(\sigma,\) on the outside surface of the shell?

A solid, nonconducting sphere of radius \(a\) has total charge \(Q\) and a uniform charge distribution. Using Gauss's Law, determine the electric field (as a vector) in the regions \(ra\) in terms of \(Q\).

Two parallel, infinite, nonconducting plates are \(10.0 \mathrm{~cm}\) apart and have charge distributions of \(+1.00 \mu \mathrm{C} / \mathrm{m}^{2}\) and \(-1.00 \mu \mathrm{C} / \mathrm{m}^{2} .\) What is the force on an electron in the space between the plates? What is the force on an electron located outside the two plates near the surface of one of the two plates?

A solid sphere of radius \(R\) has a nonuniform charge distribution \(\rho=A r^{2},\) where \(A\) is a constant. Determine the total charge, \(Q\), within the volume of the sphere.

Two parallel, uniformly charged, infinitely long wires carry opposite charges with a linear charge density \(\lambda=1.00 \mu \mathrm{C} / \mathrm{m}\) and are \(6.00 \mathrm{~cm}\) apart. What is the magnitude and direction of the electric field at a point midway between them and \(40.0 \mathrm{~cm}\) above the plane containing the two wires?

A sphere centered at the origin has a volume charge distribution of \(120 \mathrm{nC} / \mathrm{cm}^{3}\) and a radius of \(12 \mathrm{~cm}\). The sphere is centered inside a conducting spherical shell with an inner radius of \(30.0 \mathrm{~cm}\) and an outer radius of \(50.0 \mathrm{~cm}\). The charge on the spherical shell is \(-2.0 \mathrm{mC}\). What is the magnitude and direction of the electric field at each of the following distances from the origin? a) at \(r=10.0 \mathrm{~cm}\) c) at \(r=40.0 \mathrm{~cm}\) b) at \(r=20.0 \mathrm{~cm}\) d) at \(r=80.0 \mathrm{~cm}\)

A thin, hollow, metal cylinder of radius \(R\) has a surface charge distribution \(\sigma\). A long, thin wire with a linear charge density \(\lambda / 2\) runs through the center of the cylinder. Find an expression for the electric fields and the direction of the field at each of the following locations: a) \(r \leq R\) b) \(r \geq R\)

Two infinite sheets of charge are separated by \(10.0 \mathrm{~cm}\) as shown in the figure. Sheet 1 has a surface charge distribution of \(\sigma_{1}=3.00 \mu \mathrm{C} / \mathrm{m}^{2}\) and sheet 2 has a surface charge distribution of \(\sigma_{2}=-5.00 \mu \mathrm{C} / \mathrm{m}^{2}\). Find the total electric field (magnitude and direction) at each of the following locations: a) at point \(P, 6.00 \mathrm{~cm}\) to the left of sheet 1 b) at point \(P^{\prime} 6.00 \mathrm{~cm}\) to the right of sheet 1

A conducting solid sphere of radius \(20.0 \mathrm{~cm}\) is located with its center at the origin of a three-dimensional coordinate system. A charge of \(0.271 \mathrm{nC}\) is placed on the sphere. a) What is the magnitude of the electric field at point \((x, y, z)=\) \((23.1 \mathrm{~cm}, 1.1 \mathrm{~cm}, 0 \mathrm{~cm}) ?\) b) What is the angle of this electric field with the \(x\) -axis at this point? c) What is the magnitude of the electric field at point \((x, y, z)=\) \((4.1 \mathrm{~cm}, 1.1 \mathrm{~cm}, 0 \mathrm{~cm}) ?\)

A solid nonconducting sphere of radius \(a\) has a total charge \(+Q\) uniformly distributed throughout its volume. The surface of the sphere is coated with a very thin (negligible thickness) conducting layer of gold. A total charge of \(-2 Q\) is placed on this conducting layer. Use Gauss's Law to do the following. a) Find the electric field \(E(r)\) for \(ra\) (outside the coated sphere, beyond the sphere and the gold layer). c) Sketch the graph of \(E(r)\) versus \(r\). Comment on the continuity or discontinuity of the electric field, and relate this to the surface charge distribution on the gold layer.

A solid nonconducting sphere has a volume charge distribution given by \(\rho(r)=(\beta / r) \sin (\pi r / 2 R) .\) Find the total charge contained in the spherical volume and the electric field in the regions \(rR\). Show that the two expressions for the electric field equal each other at \(r=R\).

A cube has an edge length of \(1.00 \mathrm{~m} .\) An electric field acting on the cube from outside has a constant magnitude of \(150 \mathrm{~N} / \mathrm{C}\) and its direction is also constant but unspecified (not necessarily along any edges of the cube). What is the total charge within the cube?

Consider a long horizontally oriented conducting wire with \(\lambda=4.81 \cdot 10^{-12} \mathrm{C} / \mathrm{m} .\) A proton \(\left(\mathrm{mass}=1.67 \cdot 10^{-27} \mathrm{~kg}\right)\) is placed \(0.620 \mathrm{~m}\) above the wire and released. What is the magnitude of the initial acceleration of the proton?