Electric Fields

 Figure 22-22 shows three arrangements of electric field lines. In each arrangement, a proton is released from rest at point A and is then accelerated through point B by the electric field. Points A and B have equal separations in the three arrangements. Rank the arrangements according to the linear momentum of the proton at point B, greatest first.

Figure 22-23 shows two square arrays of charged particles. The squares, which are centered on point P, are misaligned. The particles are separated by either d or d/2 along the perimeters of the squares. What are the magnitude and direction of the net electric field at P?

In Fig. 22-24, two particles of charge $\mathbf{-}\mathit{q}$ are arranged symmetrically about the y axis; each produces an electric field at point P on that axis. (a) Are the magnitudes of the fields at P equal? (b) Is each electric field directed toward or away from the charge producing it? (c) Is the magnitude of the net electric field at P equal to the sum of the magnitudes E of the two field vectors (is it equal to 2E)? (d) Do the x components of those two field vectors add or cancel? (e) Do their y components add or cancel? (f) Is the direction of the net field at P that of the canceling components or the adding components? (g) What is the direction of the net field?

Figure 22-25 shows four situations in which four charged particles are evenly spaced to the left and right of a central point. The charge values are indicated. Rank the situations according to the magnitude of the net electric field at the central point, greatest first.

Figure 22-26 shows two charged particles fixed in place on an axis. (a) Where on the axis (other than at an infinite distance) is there a point at which their net electric field is zero: between the charges, to their left, or to their right? (b) Is there a point of zero net electric field anywhere off the axis (other than at an infinite distance)?

 In Fig. 22-27, two identical circular non-conducting rings are centered on the same line with their planes perpendicular to the line. Each ring has charge that is uniformly distributed along its circumference. The rings each produce electric fields at points along the line. For three situations, the charges on rings A and B are, respectively, (1) ${\mathit{q}}_{\mathbf{0}}$and ${\mathit{q}}_{\mathbf{0}}$, (2) $\mathbf{-}{\mathit{q}}_{\mathbf{0}}$and $\mathbf{-}{\mathit{q}}_{\mathbf{0}}$, and (3) $\mathbf{-}{\mathit{q}}_{\mathbf{0}}$and.${\mathit{q}}_{\mathbf{0}}$ Rank the situations according to the magnitude of the net electric field at (a) point ${\mathit{P}}_{\mathbf{1}}$midway between the rings, (b) point${\mathit{P}}_{\mathbf{2}}$ at the center of ring B, and (c) point${\mathit{P}}_{\mathbf{3}}$ to the right of ring B, greatest first.

The potential energies associated with four orientations of an electric dipole in an electric field are (1)$\mathbf{-}\mathbf{5}{\mathit{U}}_{\mathbf{0}}$ , (2) $\mathbf{-}\mathbf{7}{\mathit{U}}_{\mathbf{0}}$, (3)$\mathbf{3}{\mathit{U}}_{\mathbf{0}}$ , and (4) $\mathbf{5}{\mathit{U}}_{\mathbf{0}}$, where${\mathit{U}}_{\mathbf{0}}$  is positive. Rank the orientations according to (a) the angle between the electric dipole moment  $\overrightarrow{\mathbf{p}}$and the electric field  $\overrightarrow{\mathbf{E}}$and (b) the magnitude of the torque on the electric dipole, greatest first.

Sketch qualitatively the electric field lines both between and outside two concentric conducting spherical shells when a uniformpositive charge${\mathit{q}}_{\mathbf{1}}$ is on the inner shell and a uniform negative charge $\mathbf{-}{\mathit{q}}_{\mathbf{2}}$ is on the outer. Consider the cases,${\mathit{q}}_{\mathbf{1}}\mathbf{=}{\mathit{q}}_{\mathbf{2}}$,${\mathit{q}}_{\mathbf{1}}\mathbf{>}{\mathit{q}}_{\mathbf{2}}$ and ${\mathit{q}}_{\mathbf{1}}\mathbf{<}{\mathit{q}}_{\mathbf{2}}$.

Figure 22-28 shows two disks and a flat ring, each with the same uniform charge Q. Rank the objects according to the magnitude of the electric field they create at points P (which are at the same vertical heights), greatest first.

In Fig. 22-29, an electron e travels through a small hole in plate A and then toward plate B. A uniform electric field in the region between the plates then slows the electron without deflecting it. (a) What is the direction of the field? (b) Four other particles similarly travel through small holes in either plate A or plate B and then into the region between the plates. Three have charges$\mathbf{+}{\mathit{q}}_{\mathbf{1}}$
 ,$\mathbf{+}{\mathit{q}}_{\mathbf{2}}$
, and.$\mathbf{-}{\mathit{q}}_{\mathbf{3}}$The fourth (labeled n) is a neutron, which is electrically neutral. Does the speed of each of those four other particles increase, decrease, or remain the same in the region between the plates?

$\mathbf{+}\mathit{Q}$In Fig. 22-30a, a circular plastic rod with uniform charge $\mathbf{+}\mathit{Q}$produces an electric field of magnitude E at the center of curvature (at the origin). In Figs. 22-30b, c, and d, more circular rods, each with identical uniform charges , are added until the circle is complete. A fifth arrangement (which would be labeled e) is like that in d except the rod in the fourth quadrant has charge$\mathbf{-}\mathit{Q}$
 . Rank the five arrangements according to the magnitude of the electric field at the center of curvature, greatest first.

When three electric dipoles are near each other, they each experience the electric field of the other two, and the three-dipole system has a certain potential energy. Figure 22-31 shows two arrangements in which three electric dipoles are side by side. Each dipole has the same magnitude of electric dipole moment, and the spacing between adjacent dipoles is identical. In which arrangement is the potential energy of the three-dipole system greater?

Figure 22-32 shows three rods, each with the same charge Q spread uniformly along its length. Rods a (of length L) and b (oflength L/2) are straight, and points P are aligned with their midpoints.Rod c (of length L/2) forms a complete circle about point P. Rank the rods according to the magnitude of the electric field theycreate at points P, greatest first.

Figure 22-33 shows five protons that are launched in a uniform electric field $\overrightarrow{E}$; the magnitude and direction of the launch velocities are indicated. Rank the protons according to the magnitude of their accelerations due to the field, greatest first.

In Fig. 22-34 the electric field lines on the left have twice the separation of those on the right. (a) If the magnitude of the field at A is$\mathbf{40}\mathbf{ }\mathit{N}\mathbf{/}\mathit{C}$, what is the magnitude of the force on a proton at A? (b) What is the magnitude of the field at B?

The nucleus of a plutonium$\mathbf{-}\mathbf{239}$atom contains $\mathbf{94}\mathbf{ }\mathit{p}\mathit{r}\mathit{o}\mathit{t}\mathit{o}\mathit{n}\mathit{s}\mathbf{.}$. Assume that the nucleus is a sphere with radius $\mathbf{6}\mathbf{.}\mathbf{64}\mathbf{ }\mathbf{fm}$ and with the charge of the protons uniformly spread through the sphere. At the surface of the nucleus, what are the (a) magnitude and (b) direction (radially inward or outward) of the electric field produced by the protons?

Two charged particles are attached to an x-axis: Particle 1 of charge $\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}\mathbf{C}$ is at position $\mathbf{x}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{6}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{cm}$ and particle 2 of charge +  $\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}\mathbf{C}$ is at position. Midway between the particles, what is their net electric field in unit-vector notation?

A charged particle produces an electric field with a magnitude of $\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{ }\mathbf{N}\mathbf{/}\mathbf{C}\mathbf{ }$ at a point that is $\mathbf{ }\mathbf{50}\mathbf{ }\mathbf{cm}$ away from the particle. What is the magnitude of the particle’s charge?

What is the magnitude of a point charge that would create an electric field of $\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{N}\mathbf{/}\mathbf{C}$ at points $\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{\hspace{0.33em}}\mathbf{m}$ away?

In Fig. 22-35, the four particles form a square of edge length $\mathbf{a}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{cm}$ and have charges,${\mathbf{q}}_{\mathbf{1}}\mathbf{=}\mathbf{+}\mathbf{10}\mathbf{.}\mathbf{0}\mathbf{nC}$,${\mathbf{q}}_{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{20}\mathbf{.}\mathbf{0}\mathbf{nC}\mathbf{,}{\mathbf{q}}_{\mathbf{3}}\mathbf{=}\mathbf{-}\mathbf{20}\mathbf{.}\mathbf{0}\mathbf{nC}$, and ${\mathbf{q}}_{\mathbf{4}}\mathbf{=}\mathbf{+}\mathbf{10}\mathbf{.}\mathbf{0}\mathbf{nC}$. In unit-vector notation, what net electric field do the particles produce at the square’s center?

In Fig. 22-36, the four particles are fixed in place and have charges, ${\mathbf{q}}_{\mathbf{1}}\mathbf{=}{\mathbf{q}}_{\mathbf{2}}\mathbf{=}\mathbf{+}\mathbf{5}\mathbf{e}\mathbf{ }\mathbf{,}{\mathbf{q}}_{\mathbf{3}}\mathbf{ }\mathbf{=}\mathbf{+}\mathbf{3}\mathbf{e}\mathbf{ }\mathbf{and}\mathbf{ }{\mathbf{q}}_{\mathbf{4}}\mathbf{=}\mathbf{-}\mathbf{12}\mathbf{e}$. Distance, d = 5.0 mm. What is the magnitude of the net electric field at point P due to the particles?

Figure 22-37 shows two charged particles on an x-axis: $\mathbf{-}\mathbf{q}\mathbf{=}\mathbf{-}\mathbf{3}\mathbf{.}\mathbf{20}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{19}}\mathbf{C}$ at $\mathbf{x}\mathbf{=}\mathbf{-}\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{m}$$\mathbf{q}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{20}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{19}}$ and  at $\mathit{x}\mathbf{=}\mathbf{+}\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{m}$. What are the (a) magnitude and (b) direction (relative to the positive direction of the x-axis) of the net electric field produced at point P at $\mathbf{y}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{00}\mathbf{m}$?

Figure 22-38a shows two charged particles fixed in place on an x-axis with separation L. The ratio ${\mathit{q}}_{\mathbf{1}}\mathbf{/}{\mathit{q}}_{\mathbf{2}}$ of their charge magnitudes is . Figure 22-38b shows the x component ${\mathit{E}}_{\mathbf{n}\mathbf{e}\mathbf{t}}\mathbf{,}\mathit{X}$of their net electric field along the x-axis just to the right of particle 2. The x-axis scale is set by ${\mathit{x}}_{\mathbf{s}}\mathbf{=}\mathbf{30}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{c}\mathit{m}$. (a) At what value of $\mathit{x}\mathbf{>}\mathbf{0}\mathbf{ }\mathbf{is}\mathbf{ }{\mathit{E}}_{\mathbf{n}\mathbf{e}\mathbf{t}}\mathbf{,}\mathit{x}$  is  maximum? (b) If particle 2 has charge $\mathbf{-}{\mathit{q}}_{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{3}\mathit{e}$, what is the value of that maximum?

Two charged particles are fixed to an x axis: Particle 1of charge  ${\mathit{q}}_{\mathbf{1}}\mathbf{=}\mathbf{ }\mathbf{2}\mathbf{.1}\mathbf{ }\mathbf{\times }\mathbf{ }{\mathbf{10}}^{\mathbf{-}\mathbf{8}}\mathit{C}$is at position  $\mathit{x}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{20}\mathbf{ }\mathit{c}\mathit{m}$and particle 2 of charge ${\mathit{q}}_{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{4}\mathbf{.00}{\mathit{q}}_{\mathbf{1}}$ is at position $\mathit{x}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{70}\mathbf{ }\mathit{c}\mathit{m}$.At what coordinate on the axis (other than at infinity) is the net electric field produced by the two particles equal to zero?

Figure 22-39 shows an uneven arrangement of electrons (e) and protons (p) on a circular arc of radius r=2.00 cm, with angles , , , and . What are the (a) magnitude and (b) direction (relative to the positive direction of the x axis) of the net electric field produced at the center of the arc?

Figure 22-40 shows a proton (p) on the central axis through a disk with a uniform charge density due to excess electrons. The disk is seen from an edge-on view. Three of those electrons are shown: electron e_c at the disk center and electrons e_s at opposite sides of the disk, at radius R from the center. The proton is initially at distance z=R=2.00 cm from the disk. At that location, what are the magnitudes of (a) the electric field $\overrightarrow{{\mathbf{E}}_{\mathbf{c}}}$ due to electron e_c and (b) the net electric field ${\overrightarrow{\mathbf{E}}}_{\mathbf{s}\mathbf{,}\mathbf{net}}$ due to electrons e_s? The proton is then moved to z=R/10.0. What then are the magnitudes of (c) ${\overrightarrow{\mathbf{E}}}_{\mathbf{c}}$ and ${\overrightarrow{\mathbf{E}}}_{\mathbf{s}\mathbf{,}\mathbf{net}}$ (d)  at the proton’s location? (e) From (a) and (c) we see that as the proton gets nearer to the disk, the magnitude of ${\overrightarrow{\mathbf{E}}}_{\mathbf{c}}$ increases, as expected. Why does the magnitude of ${\overrightarrow{\mathbf{E}}}_{\mathbf{s}\mathbf{,}\mathbf{net}}$ from the two side electrons decrease, as we see from (b) and (d)?

In Fig. 22-41, particle 1 of charge q₁ = -5.00q and particle 2 of charge q₂=+2.00q are fixed to an x axis. (a) As a multiple of distance L, at what coordinate on the axis is the net electric field of the particles zero? (b)Sketch the net electric field lines between and around the particles.

In Fig. 22-42, the three particles are fixed in place and have charges q₁=q₂=+e and q₃=+2e . Distance, a=6.0 mm. What are the (a) magnitude and (b) direction of the net electric field at point P due to the particles?

Question: Figure 22-43 shows a plastic ring of radius R 50.0 cm . Two small charged beads are on the ring: Bead 1 of charge$\mathbf{ }\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{00}\mathit{\mu }\mathit{C}$  is fixed in place at the left side; bead 2 of charge  can be moved along the ring. The two beads produce a net electric field of magnitude E at the center of the ring. At what (a) positive and (b) negative value of angle $\mathit{\theta }$  should bead 2 be positioned such that$\mathit{E}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{5}}\mathbf{\text{ N/C}}$ ?

Two charged beads are on the plastic ring in Fig. 22-44a. Bead 2, which is not shown, is fixed in place on the ring, which has radius $\mathbf{R}\mathbf{ }\mathbf{=}\mathbf{60}\mathbf{.0}\mathbf{ }\mathbf{\text{\hspace{0.17em}cm}}$. Bead 1, which is not fixed in place, is initially on the x-axis at angle $\mathit{\theta }\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{0}\mathbf{°}$. It is then moved to the opposite side, at angle $\mathit{\theta }\mathbf{ }\mathbf{=}\mathbf{180}\mathbf{°}$, through the first and second quadrants of the x-y coordinate system. Figure 22-44b gives the x component of the net electric field produced at the origin by the two beads as a function of, and Fig. 22-44c gives the y component of that net electric field. The vertical axis scales are set by  ${\mathit{E}}_{\mathbf{x}\mathbf{s}}\mathbf{=}\mathbf{ }\mathbf{5.0}\mathbf{ }\mathbf{\times }{\mathbf{10}}^{\mathbf{4}}\mathbf{\text{\hspace{0.17em}N/C}}$ and ${\mathit{E}}_{\mathbf{y}\mathbf{s}}\mathbf{ }\mathbf{=}\mathbf{-}\mathbf{9.0}\mathbf{ }\mathbf{\times }\mathbf{ }{\mathbf{10}}^{\mathbf{4}}\mathbf{\text{\hspace{0.17em}N/C}}$. (a) At what angle u is bead 2 located? What are the charges of (b) bead 1 and (c) bead 2?

The electric field of an electric dipole along the dipole axis is approximated by equations. 22-8 and 22-9. If a binomial expansion is made of Eq. 22-7, what is the next term in the expression for the dipole’s electric field along the dipole axis, that is, what is ${\mathit{E}}_{\mathbf{n}\mathbf{e}\mathbf{x}\mathbf{t}}$ in the expression$\mathit{E}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}\mathbf{\pi }{\mathbf{\epsilon }}_{\mathbf{o}}}\frac{\mathbf{q}\mathbf{d}}{{\mathbf{z}}^{\mathbf{3}}}\mathbf{+}{\mathit{E}}_{\mathbf{n}\mathbf{e}\mathbf{x}\mathbf{t}}$ ?

Figure 22-45 shows an electric dipole. What are the (a) magnitude and (b) direction (relative to the positive direction of the x axis) of the dipole’s electric field at point P, located at distance, r>>d?

Equations 22-8 and 22-9 are approximations of the magnitude of the electric field of an electric dipole, at points along the dipole axis. Consider a point P on that axis at distance$\mathit{z}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{5}\mathbf{.00}\mathit{d}$  from the dipole center (d is the separation distance between the particles of the dipole). Let ${\mathit{E}}_{\mathbf{a}\mathbf{p}\mathbf{p}\mathbf{r}}$  be the magnitude of the field at point P as approximated by 22-8 and 22-9. Let ${\mathit{E}}_{\mathbf{a}\mathbf{c}\mathbf{t}}$ be the actual magnitude. What is the ratio ${\mathit{E}}_{\mathbf{a}\mathbf{p}\mathbf{p}\mathbf{r}}\mathbf{/}{\mathit{E}}_{\mathbf{a}\mathbf{c}\mathbf{t}}$?

Electric quadruple. Figure 22-46 shows a generic electric quadruple. It consists of two dipoles with dipole moments that are equal in magnitude but opposite in direction. Show that the value of E on the axis of the quadruple for a point P a distance z from its center (assume$\mathit{z}\mathbf{ }\mathbf{\gg }\mathit{d}$ ) is given by $\mathit{E}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}\mathbf{\pi }{\mathbf{\epsilon }}_{\mathbf{o}}}\frac{\mathbf{3}\mathbf{Q}}{{\mathbf{z}}^{\mathbf{4}}}$in which  is known as the quadruple moment $\mathit{Q}\mathbf{ }\mathbf{(}\mathbf{=}\mathbf{2}\mathbf{q}{\mathbf{d}}^{\mathbf{2}}\mathbf{)}$of the charge distribution.

Density, density, density. (a) A charge -300e is uniformly distributed along a circular arc of radius 4.00 cm, which subtends an angle of 40^o. What is the linear charge density along the arc? (b) A charge -300e is uniformly distributed over one face of a circular disk of 2.00 cm radius. What is the surface charge density over that face? (c) A charge -300e is uniformly distributed over the surface of a sphere of radius 2.00 cm. What is the surface charge density over that surface? (d) A charge -300e is uniformly spread through the volume of a sphere of radius 2.00 cm. What is the volume charge density in that sphere?

Figure 22-47 shows two parallel non-conducting rings with their central axes along a common line. Ring 1 has uniform charge ${\mathit{q}}_{\mathit{1}}$ and radius R; ring 2 has uniform charge ${\mathit{q}}_{\mathbf{2}}$ and the same radius R. The rings are separated by distance $\mathit{d}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{00}\mathit{R}$.The net electric field at point P on the common line, at distance R from ring 1, is zero. What is the ratio ${\mathit{q}}_{\mathbf{1}}\mathbf{/}{\mathit{q}}_{\mathbf{2}}$?

A thin non-conducting rod with a uniform distribution of positive charge Q is bent into a complete circle of radius R (Fig. 22-48). The central perpendicular axis through the ring is a z axis, with the origin at the center of the ring. What is the magnitude of the electric field due to the rod at (a)  $\mathit{z}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{0}$ and (b) $\mathit{z}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{\infty }$ ? (c) In terms of R, at what positive value of z is that magnitude maximum? (d) If  $\mathit{R}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{2.00}\mathbf{ }\mathit{c}\mathit{m}$ and $\mathbf{Q}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{4}\mathbf{.00}\mathbf{ }\mathbf{\mu C}$, what is the maximum magnitude?

Figure 22-49 shows three circular arcs centered on the origin of a coordinate system. On each arc, the uniformly distributed charge is given in terms of$\mathit{Q}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{00}\mu \mathit{C}$. The radii are given in terms of$\mathit{R}\mathbf{=}\mathbf{10}\mathbf{.}\mathbf{0}\mathit{c}\mathit{m}$. What are the (a) magnitude and (b) direction (relative to the positive x direction) of the net electric field at the origin due to the arcs?

In Fig. 22-51, two curved plastic rods, one of charge +q and the other of charge-q , form a circle of radius R=8.50 cm in an x-y plane. The x axis passes through both of the connecting points, and the charge is distributed uniformly on both rods. If q=15.0 pC, what are the (a) magnitude and (b) direction (relative to the positive direction of the x axis) of the electric field E produced at P, the center of the circle?

Charge is uniformly distributed around a ring of radius R=2.40cm, and the resulting electric field magnitude E is measured along the ring’s central axis (perpendicular to the plane of the ring). At what distance from the ring’s center is E maximum?

22-52a shows a non-conducting rod with a uniformly distributed charge Q. The rod forms a half-circle with radius R and produces an electric field of magnitude  at its center of curvature P. If the arc is collapsed to a point at distance R from P (Fig. 22-52b), by what factor is the magnitude of the electric field at P multiplied?

Figure 22-53 shows two concentric rings, of radii  R and,$\mathbf{3}\mathbf{.00}\mathit{R}$ that lie on the same plane. Point P lies on the central  Z axis, at distance $\mathit{D}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{2}\mathbf{.00}\mathit{R}$  from the center of the rings. The smaller ring has uniformly distributed charge$\mathbf{+}\mathit{Q}$. In terms of Q, what is the uniformly distributed charge on the larger ring if the net electric field at P is zero?

In Fig. 22-55, positive charge $\mathbf{q}\mathbf{ }\mathbf{=}\mathbf{7}\mathbf{.81}\mathbf{ }\mathbf{pC}$ is spread uniformly along a thin non-conducting rod of length $\mathbf{L}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{14.5}\mathbf{ }\mathbf{cm}$. What are the (a) magnitude and (b) direction (relative to the positive direction of the x axis) of the electric field produced at point P, at distance $\mathbf{R}\mathbf{ }\mathbf{=}\mathbf{6.00}\mathbf{ }\mathbf{cm}$ from the rod along its perpendicular bisector?

In Fig. 22-56, a “semi-infinite” non-conducting rod (that is, infinite in one direction only) has uniform linear charge density l. Show that the electric field   $\overrightarrow{{\mathbf{E}}_{\mathbf{p}}}$at point P makes an angle of  $\mathbf{45}\mathbf{°}$with the rod and that this result is independent of the distance R. (Hint: Separately find the component of  $\overrightarrow{{\mathbf{E}}_{\mathbf{p}}}$parallel to the rod and the component perpendicular to the rod.)

A disk of radius  $\mathbf{2}\mathbf{.5}\mathbf{ }\mathbf{\text{cm}}$has a surface charge density of $\mathbf{5}\mathbf{.3}\mathbf{ }{\mathbf{\text{μC/m}}}^{\mathbf{\text{2}}}$ on its upper face. What is the magnitude of the electric field produced by the disk at a point on its central axis at distance $\mathit{z}\mathbf{ }\mathbf{=}\mathbf{12}\mathbf{ }\mathbf{\text{cm}}$ from the disk?

At what distance along the central perpendicular axis of a uniformly charged plastic disk of radius $\mathbf{0}\mathbf{.600}\mathbf{ }\mathbf{\text{m}}$ is the magnitude of the electric field equal to one-half the magnitude of the field at the center of the surface of the disk?

A circular plastic disk with radius$\mathit{R}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{2}\mathbf{.00}\mathbf{ }\mathbf{\text{cm}}$  has a uniformly distributed charge$\mathit{Q}\mathbf{=}\mathbf{+}\mathbf{(}\mathbf{2}\mathbf{.00}\mathbf{ }\mathbf{\times }\mathbf{ }{\mathbf{10}}^{\mathbf{6}}\mathbf{)}\mathit{e}$  on one face. A circular ring of width  is centered on that face, with the center of that width at radius$\mathit{r}\mathbf{ }\mathbf{=}\mathbf{0}\mathbf{.50}\mathbf{ }\mathbf{\text{cm}}$. In coulombs, what charge is contained within the width of the ring?

Suppose you design an apparatus in which a uniformly charged disk of radius R is to produce an electric field. The field magnitude is most important along the central perpendicular axis of the disk, at a point P, at distance  $\mathbf{2}\mathbf{.00}\mathit{R}$from the disk (Fig. 22-57a). Cost analysis suggests that you switch to a ring of the same outer radius R but with inner radius $\mathit{R}\mathbf{/}\mathbf{2}\mathbf{.00}$ (Fig. 22-57b). Assume that the ring will have the same surface charge density as the original disk. If you switch to the ring, by what percentage will you decrease the electric field magnitude at P?

Figure 22-58a shows a circular disk that is uniformly charged. The central z axis is perpendicular to the disk face, with the origin at the disk. Figure 22-58b gives the magnitude of the electric field along that axis in terms of the maximum magnitude $E_m$ at the disk surface. The z axis scale is set by${\mathit{z}}_{\mathbf{s}}\mathbf{=}\mathbf{ }\mathbf{8}\mathbf{.0}\mathbf{ }\mathbf{\text{cm}}$. What is the radius of the disk?