|| What are the strength and direction of the electric field at the position indicated by the dot in FIGURE P23.36? Give your answer (a) in component form and (b) as a magnitude and angle measured cw or ccw (specify which) from the positive x-axis.

FIGURE P23.36

|| What are the strength and direction of the electric field at the position indicated by the dot in FIGURE P23.37? Give your answer (a) in component form and (b) as a magnitude and angle measured cw or ccw (specify which) from the positive x-axis.

FIGURE P23.37

|| What are the strength and direction of the electric field at the position indicated by the dot in FIGURE P23.38? Give your answer (a) in component form and (b) as a magnitude and angle measured cw or ccw (specify which) from the positive x-axis.

FIGURE P23.38

|| FIGURE P23.39 shows three charges at the corners of a square. Write the electric field at point P in component form.

FIGURE P23.39

|| A $-15\text{nC}$ charge is at $x=+2.0\text{cm}$ on the x-axis. A second charge q is located somewhere on the x-axis to the left of the origin. The electric field at $y=2.0\text{cm}$ on the y-axis is $\vec{E}=3.0\times {10}^5î\text{N}\text{/}\text{C}$. What are (a) the charge q in nC and (b) its distance from the origin?

|| FIGURE P23.41 is a cross section of two infinite lines of charge that extend out of the page. Both have linear charge density $\text{λ}$. Find an expression for the electric field strength E at height y above the midpoint between the lines.

FIGURE P23.41

||| FIGURE P23.42 is a cross section of two infinite lines of charge that extend out of the page. The linear charge densities are $\pm \text{λ}$. Find an expression for the electric field strength E at height y above the midpoint between the lines.

FIGURE P23.42

|| Derive Equation 23.11 for the field $\vec{E}{}_{\text{dipole}}$ in the plane that bisects an electric dipole.

|| CALC FIGURE P23.44 shows a thin rod of length L with total charge Q.

FIGURE P23.44

1. Find an expression for the electric field strength at point P on the axis of the rod at distance r from the center.

2. Verify that your expression has the expected behavior if $r\gg L$

3. Evaluate E at $r=3.0\text{cm}$ if $L=5.0\text{cm}$ and $Q=3.0\text{nC}$.

||| CALC FIGURE P23.45 shows a thin rod of length L with total charge Q. Find an expression for the electric field $\vec{E}$ at point P. Give your answer in component form.

FIGURE P23.45

|| Show that the on-axis electric field of a ring of charge has the expected behavior when $z\ll R$ and when $z\gg R$.

|| CALC A ring of radius R has total charge Q.

1. At what distance along the z-axis is the electric field strength a maximum?

2. What is the electric field strength at this point?

|| CALC Charge Q is uniformly distributed along a thin, flexible rod of length L. The rod is then bent into the semicircle shown in FIGURE P23.48.

FIGURE P23.48

1. Find an expression for the electric field $\vec{E}$ at the center of the semicircle.

   Hint: A small piece of arc length $\text{Δ}s$ spans a small angle $\text{Δ}\theta =\text{Δ}s\text{/}R$, where R is the radius.

2. Evaluate the field strength if $L=10\text{cm}$ and $Q=30\text{nC}$.

|| CALC A plastic rod with linear charge density $\text{λ}$ is bent into the quarter circle shown in FIGURE P23.49. We want to find the electric field at the origin.

FIGURE P23.49

1. Write expressions for the x- and y-components of the electric field at the origin due to a small piece of charge at angle $\theta$.

2. Write, but do not evaluate, definite integrals for the x- and y-components of the net electric field at the origin.

3. Evaluate the integrals and write $\vec{E}{}_{\text{net}}$ in component form.

|| An infinite plane of charge with surface charge density $3.2\mu {\text{C/m}}^2$ has a 20-cm-diameter circular hole cut out of it. What is the electric field strength directly over the center of the hole at a distance of 12 cm?

Hint: Can you create this charge distribution as a superposition of charge distributions for which you know the electric field?

|| CALC A uniform electric field’s strength is increasing with time as $E=(1.5\times {10}^4+(5.0\times {10}^{10}\text{s}{}^{-1})t)\text{N}\text{/}\text{C}$. A proton is released in the field from rest at $t=0$. What is the proton’s speed $1.0\mu s$ later?

||| INT A parallel-plate capacitor has $2.0\text{cm}\times 2.0\text{c}\text{m}$ electrodes with surface charge densities $\pm 1.0\times {10}^{-6}\text{C}\text{/}{\text{m}}^2$. A proton traveling parallel to the electrodes at $1.0\times {10}^6\text{m}\text{/}\text{s}$ enters the center of the gap between them. By what distance has the proton been deflected sideways when it reaches the far edge of the capacitor? Assume the field is uniform inside the capacitor and zero outside the capacitor.

|| INT The two parallel plates in FIGURE P23.53 are 2.0 cm apart and the electric field strength between them is $1.0\times {10}^4\text{N}\text{/}\text{C}$. An electron is launched at a $45\text{°}$ angle from the positive plate. What is the maximum initial speed $v_0$ the electron can have without hitting the negative plate?

FIGURE P23.53

|| INT An electron is launched at a $45\text{°}$ angle and a speed of $5.0\times {10}^6\text{m}\text{/}\text{s}$ from the positive plate of the parallel-plate capacitor shown in FIGURE P23.54. The electron lands 4.0 cm away.

FIGURE P23.54

1. What is the electric field strength inside the capacitor?

2. What is the smallest possible spacing between the plates?

|| INT A problem of practical interest is to make a beam of electrons turn a $90\text{°}$ corner. This can be done with the parallel-plate capacitor shown in FIGURE P23.55. An electron with kinetic energy $3.0\times {10}^{-17}\text{J}$ enters through a small hole in the bottom plate of the capacitor.

FIGURE P23.55

1. Should the bottom plate be charged positive or negative relative to the top plate if you want the electron to turn to the right? Explain.

2. What strength electric field is needed if the electron is to emerge from an exit hole 1.0 cm away from the entrance hole, traveling at right angles to its original direction?

   Hint: The difficulty of this problem depends on how you choose your coordinate system.

3. What minimum separation $d_{\text{min}}$ must the capacitor plates have?

|| INT Protons traveling in the +x-direction have kinetic energy $5.4\times {10}^{-14}\text{J}$. What uniform electric field $\vec{E}$ will stop the protons in 1.0 m?

|| CALC Your physics assignment is to figure out a way to use electricity to launch a small 6.0-cm-long plastic drink stirrer. You decide that you’ll charge the little plastic rod by rubbing it with fur, then hold it near a long, charged wire, as shown in FIGURE P23.57. When you let go, the electric force of the wire on the plastic rod will shoot it away. Suppose you can uniformly charge the plastic stirrer to 10 nC and that the linear charge density of the long wire is $1.0\times {10}^{-7}\text{C}\text{/}\text{m}$. What is the net electric force on the plastic stirrer if the end closest to the wire is 2.0 cm away?

FIGURE P23.57

Hint: The stirrer cannot be modeled as a point charge; an integration is required.

||| INT The combustion of fossil fuels produces micron-sized particles of soot, one of the major components of air pollution. The terminal speeds of these particles are extremely small, so they remain suspended in air for very long periods of time. Furthermore, very small particles almost always acquire small amounts of charge from cosmic rays and various atmospheric effects, so their motion is influenced not only by gravity but also by the earth’s weak electric field. Consider a small spherical particle of radius r, density $\rho$, and charge q. As you learned in Chapter 6, a small sphere moving with speed v experiences a drag force $F_{\text{drag}}=6\pi \eta rv$, where $\eta$ is the viscosity of the air.

1. A particle falling at its terminal speed $v_{\text{term}}$ is in equilibrium with no net force. Write Newton’s second law for this particle falling in the presence of a downward electric field of strength E, then solve to find an expression for $v_{\text{term}}$.

2. Soot is primarily carbon, and carbon in the form of graphite has a density of $2200\text{kg}\text{/}{\text{m}}^3$. In the absence of an electric field, what is the terminal speed in mm/s of a $1.0\text{-}\mu \text{m}$-diameter graphite particle? The viscosity of air at $20\text{°}$C is $1.8\times {10}^{-5}\text{kg}\text{/}\text{m}\text{s}$.

3. The earth’s electric field is typically (150 N/C, downward). In this field, what is the terminal speed in mm/s of a $1.0\text{-}\mu \text{m}$-diameter graphite particle that has acquired 250 extra electrons?

| INT In a classical model of the hydrogen atom, the electron orbits the proton in a circular orbit of radius 0.053 nm. What is the orbital frequency in rev/s? The proton is so much more massive than the electron that you can assume the proton is at rest.

|| INT A 2.0-mm-diameter glass sphere has a charge of $+1.0\text{nC}$. What speed does an electron need to orbit the sphere 1.0 mm above the surface?

|| An electric field can induce an electric dipole in a neutral atom or molecule by pushing the positive and negative charges in opposite directions. The dipole moment of an induced dipole is directly proportional to the electric field. That is, $\vec{p}=\alpha \vec{E}$, where $\alpha$ is called the polarizability of the molecule. A bigger field stretches the molecule farther and causes a larger dipole moment.

1. What are the units of $\alpha ?$

2. An ion with charge q is distance r from a molecule with polarizability $\alpha$. Find an expression for the force $\vec{E}{}_{\text{ion on dipole}}$.

|| Show that an infinite line of charge with linear charge density $\text{λ}$ exerts an attractive force on an electric dipole with magnitude $F=2\text{λ}p\text{/}4\pi {ϵ}_0{r}^2$. Assume that r, the distance from the line, is much larger than the charge separation in the dipole.

|| INT The ozone molecule $\text{O}{}_3$ has a permanent dipole moment of $1.8\times {10}^{-30}\text{C m}$. Although the molecule is very slightly bent—which is why it has a dipole moment—it can be modeled as a uniform rod of length $2.5\times {10}^{-10}\text{m}$ with the dipole moment perpendicular to the axis of the rod. Suppose an ozone molecule is in a 5000 N/C uniform electric field. In equilibrium, the dipole moment is aligned with the electric field. But if the molecule is rotated by a small angle and released, it will oscillate back and forth in simple harmonic motion. What is the frequency f of oscillation?

$(9.0\times {10}^9\text{N}{\text{m}}^2\text{/}{\text{C}}^2){\displaystyle \frac{(2.0\times {10}^{-9}\text{C})s}{(0.025\text{m}{)}^3}}=1150\text{N}\text{/}\text{C}$

$(9.0\times {10}^9\text{N}{\text{m}}^2\text{/}{\text{C}}^2){\displaystyle \frac{2(2.0\times {10}^{-7}\text{C}\text{/}\text{m})}{r}}=25\text{,}000\text{N}\text{/}\text{C}$

${\displaystyle \frac{\eta }{2{ϵ}_0}}[1-{\displaystyle \frac{z}{\sqrt{z^2+R^2}}}]={\displaystyle \frac{1}{2}}{\displaystyle \frac{\eta }{2{ϵ}_0}}$

$\begin{array}{l}2.0\times {10}^{12}\text{ m}\text{/}{\text{s}}^2={\displaystyle \frac{\left(1.60\times {10}^{-19}\text{ C}\right)E}{(1.67\times {10}^{-27}\text{ kg})}}\hfill \\ E={\displaystyle \frac{Q}{(8.85\times {10}^{-12}{\text{ C}}^2\text{/}{\text{N m}}^2){\left(0.020\text{ m}\right)}^2}}\hfill \end{array}$

||| CALC A rod of length L lies along the y-axis with its center at the origin. The rod has a nonuniform linear charge density $\text{λ}=a\left|y\right|$, where a is a constant with the units $\text{C}\text{/}{\text{m}}^2$.

1. Draw a graph of $\text{λ}$ versus y over the length of the rod.

2. Determine the constant a in terms of L and the rod’s total charge Q.

3. Find the electric field strength of the rod at distance x on the x-axis.

|||

1. CALC An infinitely long sheet of charge of width L lies in the xy-plane between $x=-L\text{/}2$ and $x=L\text{/}2$. The surface charge density is $\eta$. Derive an expression for the electric field $\vec{E}$ at height z above the centerline of the sheet.

2. Verify that your expression has the expected behavior if $z\ll L$ and if $z\gg L$.

3. Draw a graph of field strength E versus z.

|||

1. CALC An infinitely long sheet of charge of width L lies in the xy-plane between $x=-L\text{/}2$ and $x=L\text{/}2$. The surface charge density is $\eta$. Derive an expression for the electric field $\vec{E}$ along the x-axis for points outside the sheet $(x>L\text{/}2)$.

2. Verify that your expression has the expected behavior if $x\gg L$.

   Hint: $\text{ln(}1+u)\approx u\text{if u}\ll 1$.

3. Draw a graph of field strength E versus x for $x>L\text{/}2$.

||| CALC A thin cylindrical shell of radius R and length L, like a soda straw, is uniformly charged with surface charge density $\eta$. What is the electric field strength at the center of one end of the cylinder?

||| One type of ink-jet printer, called an electrostatic ink-jet printer, forms the letters by using deflecting electrodes to steer charged ink drops up and down vertically as the ink jet sweeps horizontally across the page. The ink jet forms $30\text{-}\mu \text{m}\text{-}\text{diameter}$ drops of ink, charges them by spraying 800,000 electrons on the surface, and shoots them toward the page at a speed of $20\text{m}\text{/}\text{s}$. Along the way, the drops pass through two horizontal, parallel electrodes that are 6.0 mm long, 4.0 mm wide, and spaced 1.0 mm apart. The distance from the center of the electrodes to the paper is 2.0 cm. To form the tallest letters, which have a height of 6.0 mm, the drops need to be deflected upward (or downward) by 3.0 mm. What electric field strength is needed between the electrodes to achieve this deflection? Ink, which consists of dye particles suspended in alcohol, has a density of $800\text{kg}\text{/}{\text{m}}^3$.

||| A proton orbits a long charged wire, making $1.0\times {10}^6$ revolutions per second. The radius of the orbit is 1.0 cm. What is the wire’s linear charge density?

||| You have a summer intern position with a company that designs and builds nanomachines. An engineer with the company is designing a microscopic oscillator to help keep time, and you’ve been assigned to help him analyze the design. He wants to place a negative charge at the center of a very small, positively charged metal ring. His claim is that the negative charge will undergo simple harmonic motion at a frequency determined by the amount of charge on the ring.

1. Consider a negative charge near the center of a positively charged ring centered on the z-axis. Show that there is a restoring force on the charge if it moves along the z-axis but stays close to the center of the ring. That is, show there’s a force that tries to keep the charge at $z=0$.

2. Show that for small oscillations, with $\text{amplitude}\ll R$, a particle of mass m with charge $-q$ undergoes simple harmonic motion with frequency

   $$f=\frac{1}{2\pi }\sqrt{\frac{qQ}{4\pi {ϵ}_{0}m{R}^3}}$$

   R and Q are the radius and charge of the ring.

3. Evaluate the oscillation frequency for an electron at the center of a $2.0\text{-}\mu \text{m}\text{-}\text{diameter}$ ring charged to $1.0\times {10}^{-13}\text{C}$.