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College Physics 2017

Raymond A. Serway, Chris Vuille, John hughes

Chapter 15

Electric Forces and Fields

Educators

+ 2 more educators

Problem 1

A $7.50-\mathrm{nC}$ charge is located 1.80 $\mathrm{m}$ from a $4.20-\mathrm{nC}$ charge. (a) Find the magnitude of the clectrostatic force that one particle exerts on the other. (b) Is the force attractive or repulsive?

Keshav S.
Numerade Educator

Problem 2

A charged particle $A$ exerts a force of 2.62 $\mathrm{N}$ to the right on charged particle $B$ when the particles are 13.7 $\mathrm{mm}$ apart. Particle $B$ moves straight away from $A$ to make the distance between them 17.7 $\mathrm{mm}$ . What vector force does particle $B$ then exert on $A$ ?

Eduardo M.
Numerade Educator

Problem 3

Rocket obscrvations show that dust particles in Earth's upper atmosphere are often clectrically charged. (a) Find the distance separating two dust particles if each has a charge of $+e$ and the Coulomb force between them has magnitude $1.00 \times$ $10^{-14} \mathrm{N}$ . (b) Calculate the mass of one of the dust particles if this Coulomb force would accelerate it at $4.50 \times 10^{8} \mathrm{m} / \mathrm{s}^{2} .$ (In the upper atmosphere, effects from other nearby charges typically result in a small net force and acceleration.)

Keshav S.
Numerade Educator

Problem 4

A small sphere of mass $m=7.50$ g and charge $q_{1}=32.0 \mathrm{nC}$ is attached to the end of a string and hangs vertically as in Figure $\mathrm{P} 15.4$ A second charge of equal mass and charge $q_{2}=-58.0 \mathrm{nC}$ is located below the first charge a distance $d=2.00 \mathrm{cm}$ below the first chargc as in Figure $\mathrm{P} 15.4$ . (a) Find the tension in the string. (b) If the string
can withstand a maximum tension of 0.180 $\mathrm{N}$ , what is the smallest value $d$ can have before the string breaks?

Deepak K.
Numerade Educator

Problem 5

The nucleus of $^{8} \mathrm{Be},$ which consists of 4 protons and 4 neutrons, is very unstable and spontaneously breaks into two alpha particles (helium nuclei, each consisting of 2 protons and 2 neutrons). (a) What is the force between the two alpha particles when they are $5.00 \times 10^{-15} \mathrm{m}$ apart, and (b) what is the initial magnitude of the acceleration of the alpha particles due to this force? Note that the mass of an alpha particle is 13 4.0026 $\mathrm{u} .$

Keshav S.
Numerade Educator

Problem 6

A molecule of DNA (deoxyribonucleic acid) is 2.17 mm long. The ends of the molecule become singly ionized: negative on one end, positive on the other. The helical molecule acts like a spring and compresses 1.00% upon becoming charged. Determine the effective spring constant of the molecule.

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Problem 7

Two uncharged spheres are separated by 2.00 $\mathrm{m}$ . If $3.50 \times$ $10^{12}$ electrons are removed from one sphere and placed on the other, determine the magnitude of the Coulomb force on
one of the spheres, treating the spheres as point charges.

Keshav S.
Numerade Educator

Problem 8

Four point charges are at the corners of a square of side $a$ as shown in Figure P15.8. Determine
the magnitude and direction of the resultant electric force on $q$ with $k_{c}, q,$ and a left in symbolic
form.

Deepak K.
Numerade Educator

Problem 9

Two small identical conducting spheres are placed with their centers 0.30 $\mathrm{m}$ apart. One is given a charge of $12 \times 10^{-19} \mathrm{C},$ the other a charge of $-18 \times 10^{-9} \mathrm{C}$ . (a) Find the electrostatic force exerted on one sphere by the other. (b) The spheres are connected by a conducting wire. Find the electrostatic force between the two after equilibrium is
reached, where both spheres have the same charge.

Keshav S.
Numerade Educator

Problem 10

Calculate the magnitude and direction of the Coulomb force on each of the three charges shown in Figure $\mathrm{P} 15.10$ .

Deepak K.
Numerade Educator

Problem 11

Three charges are arranged as shown in Figure P15.11. Find the magnitude and direction
of the electrostatic force on the charge at the origin.

Keshav S.
Numerade Educator

Problem 12

A positive charge $q_{1}=$ 2.70$\mu \mathrm{C}$ on a friction. less horizontal surface is attached to a spring of force contant $k$ as in Figure $\mathrm{P} 15.12$ . When a charge of $q_{2}=-8.60 \mu \mathrm{C}$ is placed 9.50 $\mathrm{cm}$ away from the positive charge, the spring stretches by $5.00 \mathrm{mm},$ reducing the distance between charges to $d=9.00 \mathrm{cm} .$ Find the value of $k$

Deepak K.
Numerade Educator

Problem 13

Three point charges are located at the corners of an cquilateral triangle as in Figure $\mathrm{P} 15.13$ . Find the magnitude and direction of the net electric force on the 2.00$\mu \mathrm{C}$ charge.

Keshav S.
Numerade Educator

Problem 14

Two identical metal blocks resting on a frictionless horizontal surface are connected by a light
metal spring having constant $k=100 \mathrm{N} / \mathrm{m}$ and unstretetehed length $L_{i}=0.400 \mathrm{m}$ as in Figure $\mathrm{P} 15.14 \mathrm{a}$ . A charge $Q$ is slowly placed on each block causing the spring to stretch to an cquilibrium length $L=0.500 \mathrm{m}$ as in Figure P15.14b. Determinc the value of $Q$ modeling the blocks as charged particles.

Lindsay E.
Numerade Educator

Problem 15

Two small metallic spheres, cach of mass $m=0.20 \mathrm{g},$ are suspended as pendulums by light strings from a common point as shown in Figure P15.15. The spheres are given the same electric
charge, and it is found that they come to equilibrium when each string is at an angle of $\theta=5.0^{\circ}$ with the vertical. If each angle of $\theta=5.0^{\circ}$ with the vertical. If each
string has length $L=30.0 \mathrm{cm},$ what is the magnitude of the charge on each sphere?

Keshav S.
Numerade Educator

Problem 16

Particle $A$ of charge $3.00 \times 10^{-4} \mathrm{C}$ is at the origin, particle $B$ of charge $-6.00 \times 10^{-4} \mathrm{C}$ is at $(4.00 \mathrm{m}, 0),$ and particle $C$ of charge $1.00 \times 10^{-4} \mathrm{C}$ is at $(0,3.00 \mathrm{m})$ , (a) What is the $x$ -component of the electric force exerted by $A$ on $C$ ? (b) What is the y-component of the force exerted by $A$ on $C^{2}(c)$ Find the magnitude of the force exerted by $B$ on $C .$ (d) Calculate the $x$ -component of the force exerted by $B$ on C. (e) Calculate the $y$ -component of the force exerted by $B$ on $C$ (f) Sum the two $x$ -componcnts to obtain the resultant $x$ -component of the electric force acting on $C$ (g) Repeat
part (f) for the $y$ -component. (h) Find the magnitude and direction of the resultant electric force acting on $C .$

Deepak K.
Numerade Educator

Problem 17

A small object of mass 3.80 $\mathrm{g}$ and charge $-18.0 \mu \mathrm{C}$ is suspended motionless above the ground when immersed in a uniformelectric field perpendicular to the ground. What is the magnitude and direction of the electric field?

Keshav S.
Numerade Educator

Problem 18

(a) Dctermine the clectric ficld strength at a point 1.00 $\mathrm{cm}$ to the left of the middle charge shown in Figure P15.10.
(b) If a charge of $-2.00 \mu \mathrm{C}$ is placed at this point, what are the magnitude and direction of the force on it?

Deepak K.
Numerade Educator

Problem 19

An electric field of magnitude $5.25 \times 10^{5} \mathrm{N} / \mathrm{C}$ points due south at a certain location. Find the magnitude and direction of the force on a $-6.00 \mu \mathrm{C}$ charge at this location.

Keshav S.
Numerade Educator

Problem 20

An electron is accelerated by a constant electric field of magnitude 300 $\mathrm{N} / \mathrm{C}$ . (a) Find the acceleration of the electron. (b) Use the equations of motion with constant acceleration to find the electron's speed after $1.00 \times 10^{-4} \mathrm{s}$ , assuming it starts from rest.

Deepak K.
Numerade Educator

Problem 21

Charge $q_{1}=1.00 \mathrm{nC}$ is at $x_{1}=0$ and charge $q_{2}=3.00 \mathrm{nC}$ is
at $x_{2}=2.00 \mathrm{m}$ . At what point betwcen the two charges is the electric ficld cqual to zero?

Keshav S.
Numerade Educator

Problem 22

A small sphere of charge $q=+68 \mu \mathrm{C}$ and mass $m=5.8 \mathrm{g}$ is attached to a light string and placed in a uniform electric field $\overrightarrow{\mathrm{E}}$ that makes an angle $\theta=37^{\circ}$ with the horizontal. The opposite end of the string is attached to a wall and the sphere is in static equilibrium when the string is horizontal as in Figure $\mathrm{P} 15.22$ . (a) Construct a free body diagram for the sphere. Find (b) the magnitucle of the electric field and (c) the ten-
sion in the string.

Deepak K.
Numerade Educator

Problem 23

A proton accelerates from rest in a uniform clectric ficld of 640 . N/C. At some later time, its speed is $1.20 \times 10^{6} \mathrm{m} / \mathrm{s}$ . (a) Find the magnitude of the acceleration of the proton. (b) How long does it take the proton to reach this speed? (c) How far has it moved in that interval? (d) What is its kinetic energy at the later time?

Keshav S.
Numerade Educator

Problem 24

(a) Find the magnitude and direction of the electric field at the position of the 2.00$\mu \mathrm{C}$ charge in Figure $\mathrm{P} 15.13$ . (b) Ilow would the electric field at that point be affected if
the charge there were doubled? Would the magnitude of the electric force be affected?

Deepak K.
Numerade Educator

Problem 25

Four point charges are located at the corners of a square. Each charge has magnitude 3.20 $\mathrm{nC}$ and the square has sides of length 2.00 $\mathrm{cm} .$ Find the magnitude of the clectric ficld at the center of the square if (a) all of the charges are positive and (b) three of the charges are positive and one is negative.

Deepak K.
Numerade Educator

Problem 26

A helium nucleus of mass $m=6.64 \times 10^{-27} \mathrm{kg}$ and charge $q=6.41 \times 10^{-19} \mathrm{C}$ is in a constant electric field of magnitude $E=2.00 \times 10^{-3} \mathrm{N} / \mathrm{C}$ pointing in the positive $x$ -direction. Neglecting other forces, calculate (a) the nucleus' acceleration and (b) its displacement after 3.00 $\mathrm{s}$ if it starts from rest.

Deepak K.
Numerade Educator

Problem 27

A charged dust particle at rest in a vacuum is held motion less by an upward-directed $475-\mathrm{N} / \mathrm{C}$ electric field. If the dus particle has a mass of $7.50 \times 10^{-10} \mathrm{kg},$ find (a) the charge or the dust particle and (b) the number of electrons that must be added to neutralize it.

Keshav S.
Numerade Educator

Problem 28

A particle of mass $1.00 \times 10^{-9} \mathrm{~kg}$ and charge $3.00 \mathrm{~pC}$ is moving in a vacuum chamber where the electric field has magnitude $2.00 \times 10^{3} \mathrm{N} / \mathrm{C}$ and is directed straight upward. Neglecting other forces except gravity, calculate the particle's (a) acceleration and (b) velocity after $2.00 \mathrm{~s}$ if it has an initial velocity of $5.00 \mathrm{~m} / \mathrm{~s}$ in the downward direction.

Deepak K.
Numerade Educator

Problem 29

Two equal positive charges are at opposite corners of a trapezoid as in Figure $\mathrm{P} 15.29$ . Find symbolicxpressions for the components of the clectric ficld at the point $P$ .

Keshav S.
Numerade Educator

Problem 30

Three point charges are located on a circular arc as shown in Figure $\mathrm{P} 15.30 .(\mathrm{a})$ What is the total electric field at $P,$ the center of the arc? (b) Find the electric force that would be
cxerted on a $-5.00$ -nC chargc placed at $P .$

Deepak K.
Numerade Educator

Problem 31

In Figure $\mathrm{P} 15.31,$ determinc the point (other than infinity) at which the total clectric ficld is zero.

Keshav S.
Numerade Educator

Problem 32

Three charges are at the corners of an equilateral triangle, as shown in Figure $\mathrm{P} 15.32$ . Calculate the electric field at a point midway between the two charges on the $x$ -axis.

Deepak K.
Numerade Educator

Problem 33

Three identical charges $(q=-5.0 \mu \mathrm{C})$ lie along a circle of radius 2.0 $\mathrm{m}$ at angles of $30^{\circ}, 150^{\circ},$ and $270^{\circ},$ as shown in Figure $\mathrm{P} 15.99$ (page 524$) .$ What is the resultant electric field at the center of the circle?

Keshav S.
Numerade Educator

Problem 34

Figure $P 15.94$ shows the electric field lines for two point charges separated by a small distance. (a) Determine the ratio $q_{1} / q_{2},$ (b) What are the signs of $q_{1}$ and $q_{2} ?$

Deepak K.
Numerade Educator

Problem 35

(a) Sketch the electric field lines around an isolated point charge $q>0$ , (b) Sketch the electric field pattern around an isolated negative point charge of magnitude - 2$q$

Keshav S.
Numerade Educator

Problem 36

(a) Sketch the electric field pattern around two positive point charges of magnitude 1$\mu \mathrm{C}$ placed close together. (b) Sketch the electric field pattern around two negative point charges of $-2 \mu \mathrm{C}$ , placed close together. (c) Sketch the pattern around two point charges of $+1 \mu \mathrm{C}$ and $-2 \mu \mathrm{C}$ , placed close together.

Deepak K.
Numerade Educator

Problem 37

Two point charges are a small distance apart. (a) Sketch the electric field lines for the two if one has a charge four times that of the other and both charges are positive. (b) Repeat for the case in which both charges are negative.

Keshav S.
Numerade Educator

Problem 38

Three equal positive charges are at the corners of an equilateral tri- angle of side a as in Figure P15.38. Assume the three charges together create an electric field. (a) Sketch the electric field lines in the plane of the charges. (b) Find the location of one point (other than `) where the electric field is zero. What are (c) the magnitude and (d) the direction of the electric field at P due to the two charges at the base?

Deepak K.
Numerade Educator

Problem 39

Refer to Figure $15.20 .$ The charge lowered into the center of the hollow conductor has a magnitude of 5$\mu \mathrm{C}$ . Find the magnitude and sign of the charge on the inside and out- side of the hollow conductor when the charge is as shown in (a) Figure $15.20 \mathrm{a},$ (b) Figure $15.20 \mathrm{b},(\mathrm{c})$ Figure $15.20 \mathrm{c},$ and (d) Figure 15.20 $\mathrm{d}$ .

Keshav S.
Numerade Educator

Problem 40

The dome of a Van de Graaff generator receives a charge of $2.0 \times 10^{-4}$ C. Find the strength of the electric field (a) inside the dome, (b) at the surface of the dome, assuming it has a radius of $1.0 \mathrm{m},$ and $(\mathrm{c}) 4.0 \mathrm{m}$ from the center of the dome. Hint: Sce Section 15.5 to revicw propertics of conductors in electrostatic equilibrium. Also, note that the points on the sur-
face are outside a spherically symmetric charge distribution; the total charge may be considered to be located at the center of the sphere.

Deepak K.
Numerade Educator

Problem 41

If the electric ficld strength in air exceeds $3.0 \times 10^{6} \mathrm{N} / \mathrm{C}$ , the air becomes a conductor. Using this fact, determine the maximum amount of charge that can be carricd by a metal sphere 2.0 $\mathrm{m}$ in radius. (See the hint in Problem 40.)

Keshav S.
Numerade Educator

Problem 42

In the Millikan oil-drop experiment illustrated in Figure 15.21 an atomizer (a sprayer with a fine nozale) is used to introduc many tiny droplets of oil between two oppositely charged parallel metal plates. Some of the droplets pick up one or more excess electrons. The charge on the plates is adjusted so that the electric force on the excess electrons exactly balances the weight of the droplet. The idea is to look for a droplet that has the smallest electric force and assume it has only one excess electron. This strategy lets the observer measure the charge on the electron. Suppose we are using an electric field of $3 \times 10^{4} \mathrm{N} / \mathrm{C}$ . The charge on onc clectron is about $1.6 \times 10^{-19}$
C. Estimate the radius of an oil drop of density 858 $\mathrm{kg} / \mathrm{m}^{3}$ for which its weight could be balanced by the clectric force of this field on one clectron. (Problem 42 is courtesy of E. F. Redish. For more problems of this type, visit www. physics.umd edu/perg/.)

Deepak K.
Numerade Educator

Problem 43

A Van de Graaff generator is charged so that a proton at its surface accclerates radially outward at $1.52 \times 10^{12} \mathrm{m} / \mathrm{s}^{2}$ . Find (a) the magnitude of the clectric force on the proton at that instant and (b) the magnitude and direction of the clectric field at the surface of the generator.

Keshav S.
Numerade Educator

Problem 44

A uniform electric field of magnitude $E=435 \mathrm{N} / \mathrm{C}$ makes an angle of $\theta=65.0^{\circ}$ with a plane surface of area $A=3.50 \mathrm{m}^{2} \mathrm{as}$
in Figure $\mathrm{P} 15.44$ . Find the electric flux through this surface.

Deepak K.
Numerade Educator

Problem 45

An electric field of intensity 3.50 kN/C is applied along the x - axis. Calculate the electric flux through a rectangular plane 0.350 m wide and 0.700 m long if (a) the plane is paral- lel to the yz - plane, (b) the plane is parallel to the xy - plane, and (c) the plane contains the y - axis and its normal makes an
angle of 40.0° with the x - axis.

Keshav S.
Numerade Educator

Problem 46

The electric field everywhere on the surface of a charged sphere of radius 0.230 m has a magnitude of 575 N/C and points radially outward from the center of the sphere. (a) What is the net charge on the sphere? (b) What can you conclude about the nature and distribution of charge inside the sphere?

Deepak K.
Numerade Educator

Problem 47

Four closed surfaces, $S_{1}$ through $S_{4},$ together with the charges $-2 Q, Q,$ and $-Q,$ are
sketched in Figure $P 15.47$ (The colored lines are the intersections of the surfaces
with the pagc.) Find the clec-tric flux through cach surface.

Keshav S.
Numerade Educator

Problem 48

A charge $q=+5.80 \mu C$ is located at the center of a regular tetrahedrom (a four-sided surface) as in Figure P15.48. Find (a) the total electric flux through the tetrahedron and (b) the clectric flux through one face of the tetrahedron.

Deepak K.
Numerade Educator

Problem 49

Figure $P 15.49$ shows a closed cylincler with cross-sectional area $A=2.00 \mathrm{m}^{2}$ . The con- stant clectric ficld $\overrightarrow{\mathrm{E}}$ has magnitude $3.50 \times 10^{3} \mathrm{N} / \mathrm{C}$ and is directed vertically upward, perpendicular to the cylinder's top and bottom surfaces so that no field lines pass through the curved surface. Calculate the electric flux
through the cylinder's (a) top and (b) bottom surfaces. (c) Determine the amount of charge inside the cylinder.

Keshav S.
Numerade Educator

Problem 50

A charge of $q=2.00 \times 10^{-9} \mathrm{C}$ is spread evenly on a thin metal disk of radius 0.200 $\mathrm{m} .$ (a) Calculate the charge density on the disk. (b) Find the magnitude of the clectric ficld
just above the center of the disk, neglecting edge effects and assuming a uniform distribution of charge.

Deepak K.
Numerade Educator

Problem 51

A point charge $q$ is located at the center of a spherical shell of radius $a$ that has a charge $-q$ uniformly distributed on its surface. Find the clectric ficld (a) for all points outside the spherical shcll and (b) for a point inside the shell a distance $r$ from the centsr.

Salamat A.
Numerade Educator

Problem 52

A charge of $1.70 \times 10^{2} \mu \mathrm{C}$ is at the center of a cube of edge 80.0 $\mathrm{cm}$ . No other charges are nearby. (a) Find the flux through the whole surface of the cube. (b) Find the flux
through cach face of the cubc. (c) Would your answers to parts (a) or (b) change if the charge were not at the center? Explain.

Deepak K.
Numerade Educator

Problem 53

Suppose the conducting spherical shell of Figure 15.29 carries a charge of 3.00 $\mathrm{nC}$ and that a charge of $-2.00 \mathrm{nC}$ is at the center of the sphere. If $a=2.00 \mathrm{m}$ and $b=2.40 \mathrm{m}$ , find the electric field at (a) $r=1.50 \mathrm{m},(\mathrm{b}) r=2.20 \mathrm{m},$ and $(\mathrm{c}) r=2.50 \mathrm{m}$ (d) What is the charge distribution on the sphere?

Keshav S.
Numerade Educator

Problem 54

A very large nonconducting plate lying in the $x$ -plane carries a charge per unit area of $\sigma .$ A second such plate located at $z=2.00 \mathrm{cm}$ and oriented parallel to the $x$ y-plane carries a charge per unit area ol $-2 \sigma .$ Find the electric lield (a) for $z<0,$ (b) $0<z<2.00 \mathrm{cm},$ and $(\mathrm{c}) z>2.00 \mathrm{cm} .$

Deepak K.
Numerade Educator

Problem 55

In deep space, two spheres each of radius 5.00 $\mathrm{m}$ are connected by a $3.00 \times 10^{2} \mathrm{m}$ nonconducting cord. If a uniformly distributed charge of 35.0 $\mathrm{mC}$ resides on the surface of each sphcre, calculate the tension in the cord.

Keshav S.
Numerade Educator

Problem 56

A nonconducting, thin plane sheet of charge carries a uniform charge per unit area of 5.20$\mu \mathrm{C} / \mathrm{m}^{2}$ as in Figure 15.30 . (a) Find the electric field at a distance of 8.70 $\mathrm{cm}$ from the plate. (b) Explain whether your result changes as the distance from the sheet is varied.

Merlin B.
Numerade Educator

Problem 57

Three point charges are aligned along the x - axis as shown in Figure P15.57. Find the electric field at the position x 5 12.0 m, y 5 0.

Keshav S.
Numerade Educator

Problem 58

A small plastic ball of mass $m=2.00 \mathrm{g}$ is suspended by a string of length $L=20.0 \mathrm{cm}$ in a uniform electric field, as shown in Figure $\mathrm{P} 15.52$ . If the ball is in equilibrium when the string makes a $\theta=15.0^{\circ}$ angle with the vertical as indicated,
what is the net charge on the ball?

Vipin B.
Numerade Educator

Problem 59

A small plastic ball of mass m 5 2.00 g is suspended by a string of length L 5 20.0 cm in a uniform electric field, as shown in Figure P15.52. If the ball is in equilibrium when the string makes a u 5 15.0° angle with the vertical as indicated, what is the net charge on the ball?

Deepak K.
Numerade Educator

Problem 60

The electrons in a particle beam each have a kinetic energy $K$ . Find the magnitude of the electric field that will stop these electrons in a distance $d$ , expressing the answer symbolically in terms of $K, e,$ and $d$ . Should the electric field point in the direction of the motion of the electron, or should it point in the opposite direction?

Deepak K.
Numerade Educator

Problem 61

A point chargc $+2 Q$ is at the origin and a point charge $-Q$ is located along the $x$ -axis
at $x=d$ as in Figure $\mathrm{P} 15.61$ . Find symbolic expressions for the components of the net
force on a third point charge $+Q$ located along the $y$ -axis at $y=d$

Keshav S.
Numerade Educator

Problem 62

A $1.00-\mathrm{g}$ cork ball having a positive charge of 2.00 $\mathrm{mC}$ is suspended vertically on a $0.500-\mathrm{m}$ -long light string in the presence of a uniform downward-directed electric field of magnitude $E=1.00 \times 10^{5} \mathrm{N} / \mathrm{C}$ as in Figure $\mathrm{P} 15.62$
If the ball is displaced slightly from the vertical, it oscillates like a simple pendulum. (a) Determine the period of the ball's oscillation. (b) Should gravity be included in the calculation for part (a)? Explain.

Deepak K.
Numerade Educator

Problem 63

Two 2.0 -g spheres are suspended by 10.0 -cm-long light strings (Fig. $\mathrm{P} 15.63 ) .$ A uni-
form electric field is applied in the $x$ -direction. If the spheres have charges of $-5.0 \times 10^{-8} \mathrm{C}$ and $+5.0 \times 10^{-8} \mathrm{C}$ , determine the electric field intensity that
enables the spheres to be in equilibrium at $\theta=10^{\circ}$ .

Keshav S.
Numerade Educator

Problem 64

A noint charge of magnitude 5.00$\mu \mathrm{C}$ is at the origin of a coordinate system, and a charge of $-4.00 \mu \mathrm{C}$ is at the point $x=$ 1.00 $\mathrm{m}$ . There is a point on the $x$ -axis, at $x$ less than infinity, where the electric field goes to zero. (a) Show by conceptual arguments hat this point cannot be located betwcen the charges. (b) Show by conceptual arguments that the point cannot be at any location between $x=0$ and negative infinity. (c) Show by conceptual arguments that the point must be between $x=1.00 \mathrm{m}$ and $x=$ positive infinity. (d) Use the values given to find the point and show that it is consistent with your conceptual argument.

Deepak K.
Numerade Educator

Problem 65

Two hard rubber spheres, each of mass $m=15.0 \mathrm{g}$ are rubbed with fur on a dry day and are then suspended with two insulating strings of length $L=$ 5.00 $\mathrm{cm}$ whose support points are a distance $d=$ 300 $\mathrm{cm}$ , from each other as shown in Figure $P 15.65$ . During the rubbing process, one sphere receives exactly twice the charge of the other. They are observed to hang at equilibrium, cach at an angle of $\theta=10.0^{\circ}$ with the vertical. Find the amount of charge on cach spherc.

Keshav S.
Numerade Educator

Problem 66

Two small beads having positive charges $q_{1}=3 q$ and $q_{2}=q$ are fixed at the opposite ends of a horizontal insulating rod of length $d=1.50 \mathrm{m}$ . The bead with charge $q_{1}$ is at the origin. As shown in Figure $\mathrm{P} 15.66,$ a third small charged bead is free to slide on the rod. At what position $x$ is the third bead in equilibrium?

Deepak K.
Numerade Educator

Problem 67

A solid conducting sphere of radius 2.00 $\mathrm{cm}$ has a charge of 8.00$\mu \mathrm{C}$ . A conducting spherical shell of inner radius 4.00 $\mathrm{cm}$ and outcr radius 5.00 $\mathrm{cm}$ is concentric with the solid sphere and has a charge of $-4.00 \mu \mathrm{C}$ . Find the clectric ficld at
(a) $r=1.00 \mathrm{cm},(\mathrm{b}) r=3.00 \mathrm{cm},(\mathrm{c}) r=4.50 \mathrm{cm},$ and $(\mathrm{d}) r=$ 7.00 $\mathrm{cm}$ from the center of this charge configuration.

Keshav S.
Numerade Educator

Problem 68

Three identical point charges, each of mass $m=0.100 \mathrm{kg}$ , hang from three strings, as shown in Figure $\mathrm{P} 15.68$ . If the lengths of the left and right strings are each $L=30.0 \mathrm{cm}$ and if the angle $\theta$ is $45.0^{\circ}$ , determine the valuc of $q$ .

Deepak K.
Numerade Educator

Problem 69

Each of the electrons in a particle beam has a kinetic energy of $1.60 \times 10^{-17} \mathrm{J}$ . (a) What is the magnitude of the uniform electric field (pointing in the direction of the electrons' move-
ment) that will stop these electrons in a distance of 10.0 $\mathrm{cm}$ ? (b) How long will it take to stop the electrons? (c) After the electrons stop, what will they do? Explain.

Keshav S.
Numerade Educator

Problem 70

Protons are projected with an initial speed $\tau_{b}=9550 \mathrm{m} / \mathrm{s}$ into a region where a uniform electric field of magnitude $E=$ 720 $\mathrm{N} / \mathrm{C}$ is present (Fig. $\mathrm{P} 15.70 ) .$ The protons are to hit a target that lies a horizontal distance of 1.27 $\mathrm{mm}$ from the point where the protons are launched. Find (a) the two projection
angles $\theta$ that will result in a hit and (b) the total duration of flight for each of the two trajectories.

Deepak K.
Numerade Educator