Chapter 15

What is the net charge of an object that has 1.0 million excess electrons?

In walking across a carpet, you acquire a net negative charge of \(50 \mu \mathrm{C}\). How many excess electrons do you have?

An alpha particle is the nucleus of a helium atom with no electrons. (a) What would be the charge on two alpha particles? (b) How many electrons would you need to add to make an alpha particle into a helium atom?

A glass rod rubbed with silk acquires a charge of \(+8.0 \times 10^{-10} \mathrm{C} .\) (a) Is the charge on the silk (1) positive, (2) zero, or (3) negative? Why? (b) What is the charge on the silk, and how many electrons have been transferred to the silk? (c) How much mass has the glass rod gained or lost?

A rubber rod rubbed with fur acquires a charge of \(-4.8 \times 10^{-9} \mathrm{C} .\) (a) Is the charge on the fur (1) positive, (2) zero, or (3) negative? Why? (b) What is the charge on the fur, and how much mass is transferred to or from the rod? (c) How much mass has the rubber rod lost or gained?

An initially uncharged electroscope is polarized by bringing a negatively charged rubber rod near the bulb. If the bulb end of the electroscope acquires a net charge of \(+2.50 \mathrm{pC}\), how many electrons are on the leaf end?

An initially neutral electrscope is charged by induction by bringing near a positively charged object. If \(3.22 \times 10^{8}\) electrons flow through the ground wire to Earth and the ground wire is then removed, what is the net charge on the electrscope?

An electron that is a certain distance from a proton is acted on by an electrical force. (a) If the electron were moved twice that distance away from the proton, would the electrical force be \((1) 2,(2) \frac{1}{2},(3) 4,\) or (4)\(\frac{1}{4}\) times the original force? Why? (b) If the initial electric force is \(F,\) and the electron were moved to one-third the original distance toward the proton, what would be the new electrical force in terms of \(F\) ?

Two identical point charges are a fixed distance apart. By what factor would the magnitude of the electric force between them change if (a) one of their charges were doubled and the other were halved, (b) both their charges were halved, and (c) one charge were halved and the other were left unchanged?

An electron and a proton are separated by \(2.0 \mathrm{nm}\). (a) What is the magnitude of the force on the electron? (b) What is the net force on the system?

Two charges originally separated by a certain distance are moved farther apart until the force between them has decreased by a factor of \(10 .\) (a) Is the new distance (1) less than \(10,\) (2) equal to \(10,\) or (3) greater than 10 times the original distance? Why? (b) If the original distance was \(30 \mathrm{~cm},\) how far apart are the charges?

Two charges are brought together until they are \(100 \mathrm{~cm}\) apart, causing the electric force between them to increase by a factor of exactly \(5 .\) What was their initial separation distance?

The distance between neighboring singly charged sodium and chlorine ions in crystals of table salt \((\mathrm{NaCl})\) is \(2.82 \times 10^{-10} \mathrm{~m}\). What is the attractive electric force between the ions?

Two charges, \(q_{1}\) and \(q_{2},\) are located at the origin and at \((0.50 \mathrm{~m}, 0),\) respectively. Where on the \(x\) -axis must a third charge, \(q_{3},\) of arbitrary sign be placed to be in electrostatic equilibrium if (a) \(q_{1}\) and \(q_{2}\) are like charges of equal magnitude, (b) \(q_{1}\) and \(q_{2}\) are unlike charges of equal magnitude, and (c) \(q_{1}=+3.0 \mu \mathrm{C}\) and \(q_{2}=-7.0 \mu \mathrm{C} ?\)

Two negative point charges are separated by \(10.0 \mathrm{~cm}\) and feel a mutual repulsive force of \(3.15 \mu \mathrm{N}\). The charge of one is three times that of the other. (a) How much charge does each have? (b) What would be the force if the total charge were instead equally distributed on both point charges?

An electron is placed on a line connecting two fixed point charges of equal charge but opposite sign. The distance between the charges is \(30.0 \mathrm{~cm}\) and the charge of each is \(4.50 \mathrm{pC}\). (a) Compute the force on the electron at \(5.0-\mathrm{cm}\) intervals starting \(5.0 \mathrm{~cm}\) from the leftmost charge and ending \(5.0 \mathrm{~cm}\) from the rightmost charge. (b) Plot the net force versus electron location using your computed values. From the plot, can you make an educated guess as to where the electron feels the least force?

If the distance from a charge is doubled, is the magnitude of the electric field (1) increased, (2) decreased, or (3) the same compared to the initial value? (b) If the original electric field due to a charge is \(1.0 \times 10^{-4} \mathrm{~N} / \mathrm{C},\) what is the magnitude of the new electric field at twice the distance from the charge?

An electron is acted on by an electric force of \(3.2 \times 10^{-14} \mathrm{~N}\). What is the magnitude of the electric field at the electron's location?

An electron is acted on by two electric forces, one of \(2.7 \times 10^{-14} \mathrm{~N}\) acting upward and a second of \(3.8 \times 10^{-14} \mathrm{~N}\) acting to the right. What is the magnitude of the electric field at the electron's location?

What are the magnitude and direction of the electric field at a point \(0.75 \mathrm{~cm}\) away from a point charge of \(+2.0 \mathrm{pC} ?\)

At what distance from a proton is the magnitude of its electric field \(1.0 \times 10^{5} \mathrm{~N} / \mathrm{C} ?\)

Two fixed charges, \(-4.0 \mu \mathrm{C}\) and \(-5.0 \mu \mathrm{C},\) are separated by a certain distance. (a) Is the net electric field at a location halfway between the two charges (1) directed toward the \(-4.0 \mu \mathrm{C}\) charge, (2) zero, or (3) directed toward the \(-5.0 \mu \mathrm{C}\) charge? Why? (b) If the charges are separated by \(20 \mathrm{~cm},\) calculate the magnitude of the net electric field halfway between the charges.

What would be the magnitude and the direction of an electric field that would just support the weight of a proton near the surface of the Earth? What about an electron?

Two charges, \(-3.0 \mu \mathrm{C}\) and \(-4.0 \mu \mathrm{C},\) are located at \((-0.50 \mathrm{~m}, 0)\) and \((0.50 \mathrm{~m}, 0),\) respectively. There is a point on the \(x\) -axis between the two charges where the electric field is zero. (a) Is that point (1) left of the origin, (2) at the origin, or (3) right of the origin? (b) Find the location of the point where the electric field is zero.

Three charges, \(+2.5 \mu \mathrm{C},-4.8 \mu \mathrm{C},\) and \(-6.3 \mu \mathrm{C},\) are located at \((-0.20 \mathrm{~m}, 0.15 \mathrm{~m}),(0.50 \mathrm{~m},-0.35 \mathrm{~m}),\) and \((-0.42 \mathrm{~m},-0.32 \mathrm{~m})\) respectively. What is the electric field at the origin?

Two charges of \(+4.0 \mu \mathrm{C}\) and \(+9.0 \mu \mathrm{C}\) are \(30 \mathrm{~cm}\) apart. Where on the line joining the charges is the electric field zero?

A particle with a mass of \(2.0 \times 10^{-5} \mathrm{~kg}\) and a charge of \(+2.0 \mu \mathrm{C}\) is released in a (parallel plate) uniform horizontal electric field of \(12 \mathrm{~N} / \mathrm{C}\). (a) How far horizontally does the particle travel in \(0.50 \mathrm{~s} ?\) (b) What is the horizontal component of its velocity at that point? (c) If the plates are \(5.0 \mathrm{~cm}\) on each side, how much charge is on each?

Two very large parallel plates are oppositely and uniformly charged. If the field between the plates is \(1.7 \times 10^{6} \mathrm{~N} / \mathrm{C},\) (a) how dense is the charge on each plate (in \(\mu \mathrm{C} / \mathrm{m}^{2}\) )? (b) How much total charge is on each plate if they are \(15.0 \mathrm{~cm}\) on a side?

Two square, oppositely charged conducting plates measure \(20 \mathrm{~cm}\) on each side. The plates are close together and parallel to each other. They each have a total charge of \(+4.0 \mathrm{nC}\) and \(-4.0 \mathrm{nC},\) respectively. (a) What is the electric field between the plates? (b) What force is exerted on an electron located between the plates? (c) What would be the electron's acceleration if it were released from rest?

A solid conducting sphere is surrounded by a thick, spherical conducting shell. Assume that a total charge \(+Q\) is placed at the center of the sphere and released. (a) After equilibrium is reached, the inner surface of the shell will have (1) negative, (2) zero, (3) positive charge. (b) In terms of \(Q\), how much charge is on the interior of the sphere? (c) The surface of the sphere? (d) The inner surface of the shell? (e) The outer surface of the shell?

A flat, triangular piece of metal with rounded corners has a net positive charge on it. Sketch the charge distribution on the surface and the electric field lines near the surface of the metal (including their direction).

Approximate a metal needle as a long cylinder with a very pointed, but slightly rounded, end. Sketch the charge distribution and outside electric field lines if the needle has an excess of electrons on it.

An electrically neutral thin, square metal slab, measuring \(5.00 \mathrm{~cm}\) on a side, is placed in a uniform external field that is perpendicular to its square area. (a) If the top of the slab becomes negatively charged, what is the direction of the external field? (b) If the external field strength is \(1250 \mathrm{~N} / \mathrm{C}\), what are the direction and strength of the field that is generated by the charges induced on the slab? (c) What is the total charge on the negative side of the slab?

Suppose a Gaussian surface encloses both a positive point charge that has six field lines leaving it and a negative point charge with twice the magnitude of charge of the positive one. What is the net number of field lines passing through the Gaussian surface?

A Gaussian surface has sixteen field lines leaving it when it surrounds a point charge of \(+10.0 \mu \mathrm{C}\) and seventy-five field lines entering it when it surrounds an unknown point charge. (a) The magnitude of the unknown charge is (1) greater than \(10.0 \mu \mathrm{C},\) (2) equal to \(10.0 \mu \mathrm{C},\) (3) less than \(10.0 \mu \mathrm{C}\). Why? (b) What is the unknown charge?

If ten field lines leave a Gaussian surface when it completely surrounds the positive end of an electric dipole, what would the count be if the surface surrounded (a) just the other end? (b) What if it surrounded both ends?

A negatively charged pith ball (mass \(6.00 \times 10^{-3} \mathrm{~g}\), charge \(-1.50 \mathrm{nC}\) ) is suspended vertically from a light nonconducting string of length \(15.5 \mathrm{~cm} .\) This apparatus is then placed in a horizontal uniform electric field. After being released, the pith ball comes to a stable position at an angle of \(12.3^{\circ}\) to the left of the vertical. (a) What is the direction of the external electric field? (b) Determine the magnitude of the electric field.

For an electric dipole, the product \(q d\) is called the dipole moment and is given the symbol \(p\). Here \(d\) is the distance between poles and \(q\) is the magnitude of the charge on either end. The dipole moment vector \(\overrightarrow{\mathbf{p}}\) has a mgnitude of \(q d\) and, by convention, points from the negative to the positive end. Assuming an electric dipole is free to move and rotate and starts from rest, (a) use a sketch to show that if it is placed in a uniform external field \(\overrightarrow{\mathbf{E}}\), it will begin to rotate so that \(\overrightarrow{\mathbf{p}}\) tries to \({ }^{\prime \prime}\) line \(u p^{\prime \prime}\) with \(\overrightarrow{\mathbf{E}}\). (b) Show that the magnitude of the torque exerted on the dipole about its center is given by \(\tau=p E \sin \theta,\) where \(\theta\) is the angle between \(\overrightarrow{\mathbf{p}}\) and \(\overrightarrow{\mathbf{E}}\). (c) What is the net force on this dipole? (d) For what angle(s) is the torque at its maximum? What about at its minimum?

A proton is fired into a uniform electric field, opposite to the direction of the field. The proton's speed upon entering the field is \(3.15 \times 10^{5} \mathrm{~m} / \mathrm{s}\), and it comes to rest \(5.25 \mathrm{~cm}\) after entering the field. (a) What is the electric field strength? (b) What is the proton's velocity when it is only \(3.50 \mathrm{~cm}\) into the field? [Hint: There is more than one answer. Why?

An electric dipole has charges of \(\pm 4.55 \mathrm{pC}\) that are separated by \(6.00 \mathrm{~cm} .\) The dipole lies on the \(x\) -axis and its center is at the origin. Located at \(y=+4.00 \mathrm{~cm}\) is a point charge carrying a charge of \(-2.50 \mathrm{pC}\). (a) Determine the net force on the dipole and its initial center of mass acceleration (including direction) if it has a total mass of \(7.25 \mathrm{ng} .\) (b) Determine the torque on the dipole about its center of mass, including direction.