||| Two point charges 2.0 cm apart have an electric potential energy $-180\mu \text{J}$. The total charge is 30 nC. What are the two charges?

||| A $-10.0\text{n}\text{C}$ point charge and a $+20.0\text{n}\text{C}$ point charge are 15.0 cm apart on the x-axis.

1. What is the electric potential at the point on the x-axis where the electric field is zero?

2. What is the magnitude of the electric field at the point on the x-axis, between the charges, where the electric potential is zero?

|| A $+3.0\text{nC}$ charge is at $x=0\text{cm}$ and a $-1.0\text{nC}$ charge is at $x=4\text{cm}$. At what point or points on the x-axis is the electric potential zero?

|| INT Two small metal cubes with masses 2.0 g and 4.0 g are tied together by a 5.0-cm-long massless string and are at rest on a frictionless surface. Each is charged to $+2.0\mu \text{C}$.

1. What is the energy of this system?

2. What is the tension in the string?

3. The string is cut. What is the speed of each cube when they are far apart?

   Hint: There are two conserved quantities. Make use of both.

|| A $-3.0\text{nC}$ charge is on the x-axis at $x=-9\text{cm}$ and a $+4.0\text{nC}$ charge is on the x-axis at $x=16\text{cm}$. At what point or points on the y-axis is the electric potential zero?

|| The four 1.0 g spheres shown in FIGURE P25.42 are released simultaneously and allowed to move away from each other. What is the speed of each sphere when they are very far apart?

FIGURE P25.42

|| A proton’s speed as it passes point 1 is $50\text{,}000\text{m}\text{/}\text{s}$. It follows the trajectory shown in FIGURE P25.43. What is the proton’s speed at point 2?

FIGURE P25.43

|| BIO Living cells “pump” singly ionized sodium ions, ${\text{Na}}^{+}$, from the inside of the cell to the outside to maintain a membrane potential $\text{Δ}{V}_{\text{membrane}}=V_{\text{in}}-V_{\text{out}}=-70\text{mV}$. It is called pumping because work must be done to move a positive ion from the negative inside of the cell to the positive outside, and it must go on continuously because sodium ions “leak” back through the cell wall by diffusion.

1. How much work must be done to move one sodium ion from the inside of the cell to the outside?

2. At rest, the human body uses energy at the rate of approximately 100 W to maintain basic metabolic functions. It has been estimated that 20% of this energy is used to operate the sodium pumps of the body. Estimate—to one significant figure—the number of sodium ions pumped per second.

|| INT An arrangement of source charges produces the electric potential $V=5000x^2$ along the x-axis, where V is in volts and x is in meters. What is the maximum speed of a 1.0 g, 10 nC charged particle that moves in this potential with turning points at $\pm 8.0\text{cm}?$

|| The electron gun in an old TV picture tube accelerates electrons between two parallel plates 1.2 cm apart with a 25 kV potential difference between them. The electrons enter through a small hole in the negative plate, accelerate, then exit through a small hole in the positive plate. Assume that the holes are small enough not to affect the electric field or potential.

1. What is the electric field strength between the plates?

2. With what speed does an electron exit the electron gun if its entry speed is close to zero?

   NOTE   The exit speed is so fast that we really need to use the theory of relativity to compute an accurate value. Your answer to part b is in the right range but a little too big.

|| A proton moves along the x-axis where some arrangement of charges has produced the potential $V\left(x\right)=V_0{}_{}\text{sin}\left(2\pi x\text{/}\text{λ}\right)$, where $V_0=5000\text{V}$ and $\text{λ}=1.0\text{mm}$.

1. What minimum speed must the proton have at $x=0$ to move down the axis without being reflected?

2. What is the maximum speed reached by a proton that at $x=0$ has the speed you calculated in part a?

|| INT A room with 3.0-m-high ceilings has a metal plate on the floor with $V=0\text{V}$ and a separate metal plate on the ceiling. A 1.0 g glass ball charged to $+4.9\text{nC}$ is shot straight up at $\text{5}\text{.0 m/s}$. How high does the ball go if the ceiling voltage is (a) $+3.0\times {10}^6\text{V}$ and (b) $-3.0\times {10}^6\text{V}?$

|| INT A 0.25 pg dust particle with 50 excess electrons is sitting at rest on top of a 5.0-cm-diameter metal sphere. Closing a switch charges the sphere almost instantaneously. To what potential must the sphere be charged to launch the dust particle to a height of 5.0 m? Ignore air resistance.

|| The electric potential in a region of space is given by $V=V_0[(x^2+2y^2)\text{/}(0.10{\text{m)}}^2]$, where $V_0$ is a constant. A proton released from rest at $(x\text{,}y)=(20\text{cm,}0\text{cm})$ reaches the origin with a speed of $7.5\times {10}^5\text{m}\text{/}\text{s}$.

1. What is $V_0$?

2. At what value of y on the y-axis should a ${\text{He}}^{+}$ ion (charge $+e$, mass 4 u) be released from rest to reach the origin with the same speed?

|| What is the escape speed of an electron launched from the surface of a 1.0-cm-diameter glass sphere that has been charged to $10\text{nC}?$

||| An electric dipole has dipole moment p. If $r\gg s$, where s is the separation between the charges, show that the electric potential of the dipole can be written

where r is the distance from the center of the dipole and $\theta$ is the angle from the dipole axis.

|| Three electrons form an equilateral triangle 1.0 nm on each side. A proton is at the center of the triangle. What is the potential energy of this group of charges?

|| INT Your lab assignment for the week is to measure the amount of charge on the 6.0-cm-diameter metal sphere of a Van de Graaff generator. To do so, you’re going to use a spring with spring constant $\text{0}\text{.65 N/m}$ to launch a small, 1.5 g bead horizontally toward the sphere. You can reliably charge the bead to 2.5 nC, and your plan is to use a video camera to measure the bead’s closest approach to the edge of the sphere as you change the compression of the spring. Your data are as follows:

Use an appropriate graph of the data to determine the sphere’s charge in nC. You can assume that the bead’s motion is entirely horizontal, that the spring is so far away that the bead has no interaction with the sphere as it’s launched, and that the approaching bead does not alter the charge distribution on the sphere.

||| A 2.0-mm-diameter glass bead is positively charged. The potential difference between a point 2.0 mm from the bead and a point 4.0 mm from the bead is 500 V. What is the charge on the bead?

| A proton is fired from far away toward the nucleus of an iron atom. Iron is element number 26, and the diameter of the nucleus is 9.0 fm. What initial speed does the proton need to just reach the surface of the nucleus? Assume the nucleus remains at rest.

| In the form of radioactive decay known as alpha decay, an unstable nucleus emits a helium-atom nucleus, which is called an alpha particle. An alpha particle contains two protons and two neutrons, thus having mass $m=4\text{u}$ and charge $q=2e$. Suppose a uranium nucleus with 92 protons decays into thorium, with 90 protons, and an alpha particle. The alpha particle is initially at rest at the surface of the thorium nucleus, which is 15 fm in diameter. What is the speed of the alpha particle when it is detected in the laboratory? Assume the thorium nucleus remains at rest.

||| A proton is fired from far away toward the nucleus of a mercury atom. Mercury is element number 80, and the diameter of the nucleus is 14.0 fm. If the proton is fired at a speed of $4.0\times {10}^7\text{m}\text{/}\text{s}$, what is its closest approach to the surface of the nucleus? Assume the nucleus remains at rest.

|| A 2.0-cm-diameter copper ring has $5.0\times {10}^9$ excess electrons. A proton is released from rest on the axis of the ring, 5.0 cm from its center. What is the proton’s speed as it passes through the center of the ring?

|| Two 10-cm-diameter electrodes 0.50 cm apart form a parallel-plate capacitor. The electrodes are attached by metal wires to the terminals of a 15 V battery. After a long time, the capacitor is disconnected from the battery but is not discharged. What are the charge on each electrode, the electric field strength inside the capacitor, and the potential difference between the electrodes

1. Right after the battery is disconnected?

2. After insulating handles are used to pull the electrodes away from each other until they are 1.0 cm apart?

3. After the original electrodes (not the modified electrodes of part b) are expanded until they are 20 cm in diameter?

|| Two 10-cm-diameter electrodes 0.50 cm apart form a parallelplate capacitor. The electrodes are attached by metal wires to the terminals of a 15 V battery. What are the charge on each electrode, the electric field strength inside the capacitor, and the potential difference between the electrodes

1. While the capacitor is attached to the battery?

2. After insulating handles are used to pull the electrodes away from each other until they are 1.0 cm apart? The electrodes remain connected to the battery during this process.

3. After the original electrodes (not the modified electrodes of part b) are expanded until they are 20 cm in diameter while remaining connected to the battery?

|| CALC Electrodes of area A are spaced distance d apart to form a parallel-plate capacitor. The electrodes are charged to $\pm q$.

1. What is the infinitesimal increase in electric potential energy dU if an infinitesimal amount of charge dq is moved from the negative electrode to the positive electrode?

2. An uncharged capacitor can be charged to $\pm Q$ by transferring charge dq over and over and over. Use your answer to part a to show that the potential energy of a capacitor charged to $\pm Q$ is $U_{\text{cap}}=\frac{1}{2}Q\text{Δ}{V}_{\text{C}}$.

|| INT We’ve seen that molecular bonds can be modeled as springs. Suppose a positive ion $(+e)$ and a negative ion $(-e)$ are attached to the ends of a microscopic spring that has an unstretched length of 0.25 nm. As the ions vibrate back and forth, their separation oscillates between 0.20 nm and 0.25 nm. What is the spring constant?

|| The potential 1.0 cm from the surface of a metal sphere is 8000 V. The potential 3.0 cm from the surface is 4000 V. What are (a) the radius of the sphere and (b) the charge on the sphere?

|| Two spherical drops of mercury each have a charge of 0.10 nC and a potential of 300 V at the surface. The two drops merge to form a single drop. What is the potential at the surface of the new drop?

| A Van de Graaff generator is a device for generating a large electric potential by building up charge on a hollow metal sphere. A typical classroom-demonstration model has a diameter of 30 cm.

1. How much charge is needed on the sphere for its potential to be 500,000 V?

2. What is the electric field strength just outside the surface of the sphere when it is charged to 500,000 V?

|| FIGURE P25.67 shows two uniformly charged spheres. What is the potential difference between points 1 and 2? Which point is at the higher potential?

FIGURE P25.67

Hint: The potential at any point is the superposition of the potentials due to all charges.

||| Two metal objects that are in contact must be at the same potential, an assertion we’ll prove in the next chapter. Suppose a metal sphere of radius R is charged to 1000 V and a second metal sphere of radius 2R is charged to 2000 V. The two spheres are brought into contact and then separated. Afterward, what is the potential of each sphere?

|| The arrangement of charges shown in FIGURE P25.69 is called a linear electric quadrupole. The positive charges are located at $y=\pm s$. Notice that the net charge is zero. Find an expression for the electric potential on the y-axis at distances $y\gg s$. Give your answer in terms of the quadrupole moment, $Q=2q{s}^2$.

FIGURE P25.69

|| CALC FIGURE P25.70 shows a thin rod of length L and charge Q. Find an expression for the electric potential a distance x away from the center of the rod on the axis of the rod.

FIGURE P25.70

||| CALC Figure P25.70 shows a thin rod of length L and charge Q. Find an expression for the electric potential a distance z away from the center of rod on the line that bisects the rod.

| FIGURE P25.72 shows a thin rod with charge Q that has been bent into a semicircle of radius R. Find an expression for the electric potential at the center.

FIGURE P25.72

|| CALC A disk with a hole has inner radius $R_{\text{in}}$ and outer radius $R_{\text{out}}$. The disk is uniformly charged with total charge Q. Find an expression for the on-axis electric potential at distance z from the center of the disk. Verify that your expression has the correct behavior when $R_{\text{in}}\to 0$.

|| CALC The wire in FIGURE P25.74 has linear charge density $\text{λ}$. What is the electric potential at the center of the semicircle?

FIGURE P25.74

$\frac{\left(9.0\times {10}^9{\text{Nm}}^2{\text{/C}}^2\right)q_1{q}_2}{0.030\text{m}}=90\times {10}^{-6}\text{J}$

$q_1+q_2=40\text{nC}$

$\frac{1}{2}(1.67\times {10}^{-27}\text{kg}){(2.5\times {10}^6\text{m/s})}^2+0=$

$\frac{1}{2}(1.67\times {10}^{-27}\text{kg})v_{\text{i}}^2+$

$\frac{(9.0\times {10}^9{\text{Nm}}^2{\text{/C}}^2)(2.0\times {10}^{-9}\text{C})(1.60\times {10}^{-19}\text{C})}{0.0010\text{m}}$

$\begin{array}{l}{\displaystyle \frac{(9.0\times {10}^9{\text{Nm}}^2{\text{/C}}^2)(3.0\times {10}^{-9}\text{\hspace{0.17em}C})}{0.030\text{m}}}+\\ {\displaystyle \frac{(9.0\times {10}^9{\text{Nm}}^2{\text{/C}}^2)(3.0\times {10}^{-9}\text{\hspace{0.17em}C})}{(0.030\text{m})+d}}=1200\text{V}\end{array}$

||| An electric dipole consists of 1.0 g spheres charged to $\pm 2.0\text{nC}$ at the ends of a 10-cm-long massless rod. The dipole rotates on a frictionless pivot at its center. The dipole is held perpendicular to a uniform electric field with field strength $1000\text{V}\text{/}\text{m}$, then released. What is the dipole’s angular velocity at the instant it is aligned with the electric field?

||| A proton and an alpha particle $(q=+2e\text{,}m=4\text{u})$ are fired directly toward each other from far away, each with an initial speed of 0.010c. What is their distance of closest approach, as measured between their centers?

||| Bead A has a mass of 15 g and a charge of $-5.0\text{nC}$. Bead B has a mass of 25 g and a charge of $-10.0\text{nC}$. The beads are held 12 cm apart (measured between their centers) and released. What maximum speed is achieved by each bead?

||| Two 2.0-mm-diameter beads, C and D, are 10 mm apart, measured between their centers. Bead C has mass 1.0 g and charge 2.0 nC. Bead D has mass 2.0 g and charge $-1.0\text{nC}$. If the beads are released from rest, what are the speeds $v_{\text{C}}$ and $v_{\text{D}}$ at the instant the beads collide?

||| CALC A thin rod of length L and total charge Q has the nonuniform linear charge distribution $\text{λ}\left(x\right)=\text{λ}{}_{0}x\text{/}L$, where x is measured from the rod’s left end.

1. What is ${\text{λ}}_0$ in terms of Q and L?

2. What is the electric potential on the axis at distance d left of the rod’s left end?

||| CALC A hollow cylindrical shell of length L and radius R has charge Q uniformly distributed along its length. What is the electric potential at the center of the cylinder?