|| INT Two 1.0 g spheres are charged equally and placed 2.0 cm apart. When released, they begin to accelerate at $150\text{m}\text{/}{\text{s}}^2$. What is the magnitude of the charge on each sphere?

|| INT The nucleus of a ${}^{125}\text{Xe}$ atom (an isotope of the element xenon with mass 125 u) is 6.0 fm in diameter. It has 54 protons and charge $q=+54e$.

1. What is the electric force on a proton 2.0 fm from the surface of the nucleus?

   Hint: Treat the spherical nucleus as a point charge.

2. What is the proton’s acceleration?

|| INT A 3.00-cm-long spring has a small plastic bead glued to each end. Charging each bead to $-25\text{ nC}$ expands the spring by 0.50 cm. What is the value of the spring constant?

|| Two 1.5 g ball bearings are 2.0 cm apart. How many electrons must be transferred from one to the other so that, when released, they each initially accelerate toward the other with $a=g?$

| A smart phone charger delivers charge to the phone, in the form of electrons, at a rate of $-0.75\text{ C/s}$. How many electrons are delivered to the phone during 30 min of charging?

|| INT Objects A and B are both positively charged. Both have a mass of 100 g, but A has twice the charge of B. When A and B are placed 10 cm apart, B experiences an electric force of 0.45 N.

1. What is the charge on A?

2. If the objects are released, what is the initial acceleration of A?

|| What is the force $\vec{F}$ on the 5.0 nC charge in FIGURE P22.43? Give your answer as a magnitude and an angle measured cw or ccw (specify which) from the $-x\text{-}\text{axis}$.

FIGURE P22.43

|| What is the force $\vec{F}$ on the 8.0 nC charge in FIGURE P22.44? Give your answer as a magnitude and an angle measured cw or ccw (specify which) from the $+x\text{-}\text{axis}$.

FIGURE P22.44

|| What is the force $\vec{F}$ on the $-1.0\text{ nC}$ charge in the middle of FIGURE P22.45 due to the four other charges? Give your answer in component form.

FIGURE P22.45

|| What is the force $\vec{F}$ on the 5.0 nC charge in FIGURE P22.46? Give your answer in component form.

FIGURE P22.46

|| What is the force $\vec{F}$ on the 1.0 nC charge at the bottom in FIGURE P22.47? Give your answer in component form.

FIGURE P22.47

|| What is the force $\vec{F}$ on the 1.0 nC charge at the bottom in FIGURE P22.48? Give your answer in component form.

FIGURE P22.48

|| A $+2.0\text{nC}$ charge is at the origin and a $-4.0\text{nC}$ charge is at $x=1.0\text{cm}$.

1. At what $x$-coordinate could you place a proton so that it would experience no net force?

2. Would the net force be zero for an electron placed at the same position? Explain.

|| Charge $q_2$ in FIGURE P22.50 is in equilibrium. What is $q_1?$

FIGURE P22.50

|| The $-2.0\text{nC}$ charge in FIGURE P22.51 is in equilibrium. What is $q$?

FIGURE P22.51

| FIGURE P22.52 shows three charges and the net force on charge $-q$. Charge $Q$ is some multiple $\alpha$ of $q$. What is $\alpha ?$

FIGURE P22.52

|| FIGURE P22.53 shows four charges at the corners of a square of side $L$. What is the magnitude of the net force on $q$?

FIGURE P22.53

|| A positive point charge $Q$ is located at $x=a$ and a negative point charge $-Q$ is at $x=-a$. A positive charge $q$ can be placed anywhere on the $y$-axis. Find an expression for $(F_{\text{net}}){}_x$, the $x$-component of the net force on $q$, as a function of $y$.

|| INT In a simple model of the hydrogen atom, the electron moves in a circular orbit of radius 0.053 nm around a stationary proton. How many revolutions per second does the electron make?

||| INT Suppose the magnitude of the proton charge differs from the magnitude of the electron charge by a mere 1 part in ${10}^9$.

1. What would be the force between two 2.0-mm-diameter copper spheres 1.0 cm apart? Assume that each copper atom has an equal number of electrons and protons.

2. Would this amount of force be detectable? What can you conclude from the fact that no such forces are observed?

||| You have two small, 2.0 g balls that have been given equal but opposite charges, but you don’t know the magnitude of the charge. To find out, you place the balls distance $d$ apart on a slippery horizontal surface, release them, and use a motion detector to measure the initial acceleration of one of the balls toward the other. After repeating this for several different separation distances, your data are as follows:

Use an appropriate graph of the data to determine the magnitude of the charge.

|| INT A 2.0 g metal cube and a 4.0 g metal cube are 6.0 cm apart, measured between their centers, on a horizontal surface. For both, the coefficient of static friction is 0.65. Both cubes, initially neutral, are charged at a rate of $7.0\text{nC/s}$. How long after charging begins does one cube begin to slide away? Which cube moves first?

|| CALC Two equal point charges 2.5 cm apart, both initially neutral, are being charged at the rate of $5.0\text{nC/s}$. At what rate (N/s) is the force between them increasing 1.0 s after charging begins?

||| You have a lightweight spring whose unstretched length is 4.0 cm. First, you attach one end of the spring to the ceiling and hang a 1.0 g mass from it. This stretches the spring to a length of 5.0 cm. You then attach two small plastic beads to the opposite ends of the spring, lay the spring on a frictionless table, and give each plastic bead the same charge. This stretches the spring to a length of 4.5 cm. What is the magnitude of the charge (in nC) on each bead?

| INT An electric dipole consists of two opposite charges $\pm q$ separated by a small distance $s$. The product $p=qs$ is called the dipole moment. FIGURE P22.61 shows an electric dipole perpendicular to an electric field $\vec{E}$. Find an expression in terms of $p$ and $E$ for the magnitude of the torque that the electric field exerts on the dipole.

FIGURE P22.61

|| INT You sometimes create a spark when you touch a doorknob after shuffling your feet on a carpet. Why? The air always has a few free electrons that have been kicked out of atoms by cosmic rays. If an electric field is present, a free electron is accelerated until it collides with an air molecule. Most such collisions are elastic, so the electron collides, accelerates, collides, accelerates, and so on, gradually gaining speed. But if the electron’s kinetic energy just before a collision is $2.0\times {10}^{-18}\text{J}$ or more, it has sufficient energy to kick an electron out of the molecule it hits. Where there was one free electron, now there are two! Each of these can then accelerate, hit a molecule, and kick out another electron. Then there will be four free electrons. In other words, as FIGURE P22.62 shows, a sufficiently strong electric field causes a “chain reaction” of electron production. This is called a breakdown of the air. The current of moving electrons is what gives you the shock, and a spark is generated when the electrons recombine with the positive ions and give off excess energy as a burst of light.

FIGURE P22.62

1. The average distance between ionizing collisions is $2.0\mu \text{m}$. (The electron’s mean free path is less than this, but most collisions are elastic collisions in which the electron bounces with no loss of energy.) What acceleration must an electron have to gain $2.0\times {10}^{-18}\text{J}$ of kinetic energy in this distance?

2. What force must act on an electron to give it the acceleration found in part a?

3. What strength electric field will exert this much force on an electron? This is the breakdown field strength. Note: The measured breakdown field strength is a little less than your calculated value because our model of the process is a bit too simple. Even so, your calculated value is close.

4. Suppose a free electron in air is 1.0 cm away from a point charge. What minimum charge is needed to cause a breakdown and create a spark as the electron moves toward the point charge?

|| INT Two 5.0 g point charges on 1.0-m-long threads repel each other after being charged to $+100\text{nC,}$ as shown in FIGURE P22.63. What is the angle $\theta ?$ You can assume that $\theta$ is a small angle.

FIGURE P22.63

|| The identical small spheres shown in FIGURE P22.64 are charged to $+100\text{nC}$ and $-100\text{nC}$. They hang as shown in a $100\text{,}000\text{N}\text{/}\text{C}$ electric field. What is the mass of each sphere?

FIGURE P22.64

|| A 10.0 nC charge is located at position $\left(x\text{,}y\right)=\left(1.0\text{ cm, }2.0\text{ cm}\right)$. At what $\left(x\text{,}y\right)$ position(s) is the electric field

1. $-225\text{,}000î\text{N}\text{/}\text{C}?$

2. $(161\text{,}000î+80\text{,}500ĵ)\text{N/C}?$

3. $(21\text{,}600î-28\text{,}800ĵ)\text{N}\text{/}\text{C}?$

| Three 1.0 nC charges are placed as shown in FIGURE P22.66. Each of these charges creates an electric field $\vec{E}$ at a point 3.0 cm in front of the middle charge.

FIGURE P22.66

1. What are the three fields $\vec{E}{}_1\text{,}\vec{E}{}_2$, and $\vec{E}{}_3$ created by the three charges? Write your answer for each as a vector in component form.

2. Do you think that electric fields obey a principle of superposition? That is, is there a “net field” at this point given by $\vec{E}{}_{\text{net}}=\vec{E}{}_1+\vec{E}{}_2+\vec{E}{}_3?$ Use what you learned in this chapter and previously in our study of forces to argue why this is or is not true.

3. If it is true, what is $\vec{E}{}_{\text{net}}?$

| INT An electric field $\vec{E}=100\text{,}000î\text{N}\text{/}\text{C}$ causes the 5.0 g point charge in FIGURE P22.67 to hang at a $20\text{°}$ angle. What is the charge on the ball?

FIGURE P22.67

|| INT An electric field $\vec{E}=200\text{,}000î\text{N}\text{/}\text{C}$ causes the point charge in FIGURE P22.68 to hang at an angle. What is $\theta ?$

FIGURE P22.68

|| INT One 1.0 g bead is charged to $+10\text{ nC}$, a second to $-10\text{ nC}$. They rotate about their common center of mass at 150 rpm. What is the distance between the beads?

|| INT

1. Starting from rest, how long does it take an electron to move 1.0 cm in a steady electric field of magnitude 100 N/C?

2. What is the electron’s speed after traveling 1.0 cm?

|| INT A 5.0 g ball charged to $1.5\mu \text{C}$ is tied to a 25-cm-long string. It swings at 250 rpm in a horizontal circle around a stationary ball charged to $-2.5\mu \text{C}$. What is the tension in the string?

${\displaystyle \frac{(9.0\times {10}^9\text{N}{\text{m}}^2\text{/}{\text{C}}^2)\times N\times (1.60\times {10}^{-19}\text{C})}{(1.0\times {10}^{-6}{\text{m)}}^2}}=1.5\times {10}^6\text{N}\text{/}\text{C}$

${\displaystyle \frac{(9.0\times {10}^9\text{N}{\text{m}}^2\text{/}{\text{C}}^2)q^2}{(0.0150{\text{m)}}^2}}=0.020\text{N}$

${\displaystyle \frac{(9.0\times {10}^9\text{N}{\text{m}}^2\text{/}{\text{C}}^2)(15\times {10}^{-9}\text{C})}{r^2}}=54\text{,}000\text{N}\text{/}\text{C}$

$\begin{array}{lll}{{\displaystyle \text{∑}}}^{\text{}}{F}_x\hfill & =\hfill & 2\times {\displaystyle \frac{(9.0\times {10}^9{\text{ N m}}^2{\text{/C}}^2)(1.0\times {10}^{-9}\text{ C})q}{{((0.020\text{ m})\text{/sin }30\text{°})}^2}}\times \text{cos }30\text{°}\hfill \\ \hfill & =\hfill & 5.0\times {10}^{-5}\text{ N}\hfill \\ {{\displaystyle \text{∑}}}^{\text{}}{F}_y\hfill & =\hfill & 0\text{ N}\hfill \end{array}$

||| Two 3.0 g point charges on 1.0-m-long threads repel each other after being equally charged, as shown in FIGURE CP22.76. What is the magnitude of the charge $q$?

FIGURE CP22.76

||| Space explorers discover an $8.7\times {10}^{17}\text{kg}$ asteroid that happens to have a positive charge of 4400 C. They would like to place their $3.3\times {10}^5\text{kg}$ spaceship in orbit around the asteroid. Interestingly, the solar wind has given their spaceship a charge of $-1.2\text{C}$. What speed must their spaceship have to achieve a 7500-km-diameter circular orbit?

||| A small 1.0 g block charged to 75 nC is placed on a 30° inclined plane. The coefficients of static and kinetic friction are 0.20 and 0.10, respectively. What minimum strength horizontal electric field is needed to keep the block from sliding down the plane?

||| The force on the $-1.0\text{nC}$ charge is as shown in FIGURE CP22.79. What is the magnitude of this force?

FIGURE CP22.79

||| In Section 22.3 we claimed that a charged object exerts a net attractive force on an electric dipole. Let’s investigate this. FIGURE CP22.80 shows a permanent electric dipole consisting of charges $+q$ and $-q$ separated by the fixed distance $s$. Charge $+Q$ is distance $r$ from the center of the dipole. We’ll assume, as is usually the case in practice, that $s\ll r$.

FIGURE CP22.80

1. Write an expression for the net force exerted on the dipole by charge $+Q$.

2. Is this force toward $+Q$ or away from $+Q?$ Explain.

3. Use the binomial approximation ${(1+x)}^{-n}\approx 1-nx$ if $x\ll 1$ to show that your expression from part a can be written $F_{\text{net}}=2KqQs\text{/}{r}^3$.

4. How can an electric force have an inverse-cube dependence? Doesn’t Coulomb’s law say that the electric force depends on the inverse square of the distance? Explain.