what equal positive charges would have to be placed on earth and on the moon
Problem 1
(I) What is the magnitude of the electric force of attraction between an iron nucleus $( q = + 26 due east )$ and its innermost electron if the distance between them is $1.five \times 10 ^ { - 12 } \mathrm { grand } ?$
Shital R.
Numerade Educator
Problem ii
(I) How many electrons make upwards a charge of $- 38.0 \mu \mathrm { C } ?$
Ceren U.
Texas Tech University
Trouble 3
(1) What is the magnitude of the force a $+ 25 \mu \mathrm { C }$ accuse exerts on a $+ 2.5 \mathrm { mC }$ accuse 28$\mathrm { cm }$ abroad?
Shital R.
Numerade Educator
Trouble 4
(I) What is the repulsive electrical force between two protons $4.0 \times 10 ^ { - fifteen } \mathrm { m }$ apart from each other in an diminutive nucleus?
Ceren U.
Texas Tech University
Trouble v
(2) When an object such every bit a plastic rummage is charged past rubbing information technology with a cloth, the net charge is typically a few microcoulombs. If that accuse is $iii.0 \mu C ,$ by what percentage does the mass of a $35 -$ g rummage alter during charging?
Manish M.
Numerade Educator
Problem 6
(II) Two charged dust particles exert a forcefulness of $3.2 \times 10 ^ { - ii } \mathrm { North }$ on each other. What will be the force if they are moved so they are only one-eighth as far apart?
Ceren U.
Texas Tech University
Trouble 7
(II) Two charged spheres are 8.45$\mathrm { cm }$ apart. They are moved, and the force on each of them is establish to accept been tripled. How far apart are they now?
Manish K.
Numerade Educator
Problem viii
(II) A person scuffing her anxiety on a wool rug on a dry day accumulates a net charge of - 46 $\mu \mathrm { C }$ . How many excess electrons does she go, and by how much does her mass increase?
Ceren U.
Texas Tech University
Trouble 9
(II) What is the total charge of all the electrons in a xv -kg bar of gilt? What is the net accuse of the bar? (Gilded has 79 electrons per atom and an atomic mass of 197 u.)
Manne A.
Numerade Educator
Problem 10
(Two) Compare the electric forcefulness holding the electron in orbit $\left( r = 0.53 \times 10 ^ { - ten } \mathrm { m } \right)$ around the proton nucleus of the hydrogen atom, with the gravitational force betwixt the same electron and proton. What is the ratio of these two forces?
Ceren U.
Texas Tech University
Problem eleven
(II) 2 positive signal charges are a fixed distance apart. The sum of their charges is $Q _ { \mathrm { T } }$ . What charge must each have in order to (a) maximize the electric force betwixt them, and $( b )$ minimize it?
Shital R.
Numerade Educator
Trouble 12
(Ii) Particles of accuse $+ 75 , + 48 ,$ and $- 85 \mu$ Intendance placed in a line (Fig. $52 ) .$ The center one is 0.35 from each of the others. Calculate the internet force on each charge due to the other two.
Suzanne W.
Numerade Educator
Problem 13
(Ii) Three charged particles are placed at the corners of an equilateral triangle of side 1.20$\mathrm { thou }$ (Fig. $53 ) .$ The charges are $+ seven.0 \mu \mathrm { C } , - eight.0 \mu \mathrm { C } ,$ and $- 6.0 \mu \mathrm { C }$ Calculate the magnitude and direction of the internet forcefulness on each due to the other two.
Pritesh R.
Numerade Educator
Trouble 14
(Ii) Two small nonconducting spheres have a total charge of ninety.0$\mu \mathrm { C }$ (a) When placed 1.xvi$\mathrm { m }$ apart, the force each exerts on the other is 12.0$\mathrm { Due north }$ and is repulsive. What is the charge on each? (b) What if the force were attractive?
Ceren U.
Texas Tech University
Problem 15
(II) A charge of 4.15$\mathrm { mC }$ is placed at each corner of a foursquare 0.100$\mathrm { k }$ on a side. Determine the magnitude and direction of the force on each accuse.
Darshan Yard.
Numerade Educator
Problem sixteen
(Ii) Two negative and two positive point charges (magnitude $Q = 4.15 \mathrm { mC }$ ) are placed on opposite corners of a square as shown in Fig. $54 .$ Decide the magnitude and management of the force on each accuse.
Ceren U.
Texas Tech Academy
Problem 17
(II) A charge $Q$ is transferred from an initially uncharged plastic ball to an identical ball 12$\mathrm { cm }$ away. The force of allure is then 17$\mathrm { mN }$ . How many electrons were transferred from 1 ball to the other?
Shital R.
Numerade Educator
Trouble 18
(Iii) Two charges, $- Q _ { 0 }$ and $- 4 Q _ { 0 } ,$ are a altitude $\ell$ autonomously. These 2 charges are free to motility merely practice not because there is a third charge nearby. What must be the magnitude of the 3rd charge and its placement in guild for the first 2 to be in equilibrium?
Ceren U.
Texas Tech Academy
Problem 19
(3) 2 positive charges $+ Q$ are affixed rigidly to the $x$ axis, 1 at $x = + d$ and the other at $x = - d .$ A third accuse $+ q$ of mass $1000 ,$ which is constrained to movement only along the $x$ axis, is displaced from the origin past a modest distance $s \ll d$ and so released from rest. $( a )$ Prove that (to a good approximation) $+ q$ will execute simple harmonic motion and determine an expression for its oscillation period $T .$ (b) If these three charges are each singly ionized sodium atoms $( q = Q = + east )$ at the equilibrium spacing $d = 3 \times 10 ^ { - x } \mathrm { m }$ typical of the atomic spacing in a solid, notice $T$ in picoseconds.
Linda W.
Numerade Educator
Problem 20
(Three) Two small charged spheres hang from cords of equal length $\ell$ as shown in Fig. 55 and make small angles $\theta _ { i }$ and $\theta _ { 2 }$ with the vertical. $( a )$ If $Q _ { 1 } = Q , \quad Q _ { 2 } = 2 Q ,$ and $m _ { i } = m _ { ii } = g ,$ make up one's mind the ratio $\theta _ { 1 } / \theta _ { 2 } .$ and If $Q _ { i } = Q , \quad Q _ { 2 } = two Q , \quad yard _ { 1 } = m , \quad$ and $m _ { 2 } = two m ,$ decide the ratio $\theta _ { 1 } / \theta _ { ii } . ( c )$ Esti- mate the altitude between the spheres for each case.
Ceren U.
Texas Tech University
Problem 21
(I) What are the magnitude and direction of the electric force on an electron in a uniform electric field of strength 1920$\mathrm { N } / \mathrm { C }$ that points due east?
Shital R.
Numerade Educator
Problem 22
(I) A proton is released in a uniform electric field, and it experiences an electric forcefulness of $two.18 \times 10 ^ { - 14 } \mathrm { N }$ toward the s. What are the magnitude and direction of the electric field?
Ceren U.
Texas Tech University
Problem 23
(I) Determine the magnitude and management of the electric field xvi.4$\mathrm { cm }$ directly above an isolated $33.0 \times 10 ^ { - 6 }$ Ccharge.
Shital R.
Numerade Educator
Problem 24
(I) A downwards electric force of 8.4$\mathrm { N }$ is exerted on a $- 8.8 \mu \mathrm { C }$ accuse. What are the magnitude and direction of the electrical field at the position of this charge?
Ceren U.
Texas Tech Academy
Problem 25
(I) The electric force on a $+ 4.20 - \mu \mathrm { C }$ charge is $\left( 7.22 \times ten ^ { - iv } \mathrm { Due north } \right) \hat { \mathrm { j } }$ . What is the electric field at the position of the charge?
Shital R.
Numerade Educator
Problem 26
(I) What is the electric field at a point when the force on a $1.25 - \mu \mathrm { C }$ accuse placed at that point is $\vec { \mathbf { F } } = ( 3.0 \hat { \mathbf { i } } - 3.9 \hat { \mathbf { j } } ) \times 10 ^ { - 3 } \mathbf { North } ?$
Ceren U.
Texas Tech University
Problem 27
(Ii) Make up one's mind the magnitude of the acceleration experienced by an electron in an electric field of 576$\mathrm { North } / \mathrm { C }$ . How does the direction of the dispatch depend on the management of thefield at that point?
Shital R.
Numerade Educator
Trouble 28
(Two) Determine the magnitude and direction of the electrical field at a point midway between a $- viii.0 \mu \mathrm { C }$ and a $+ 5.eight \mu \mathrm { C }$ charge 8.0$\mathrm { cm }$ apart. Assume no other charges are nearby.
Ceren U.
Texas Tech University
Problem 29
(Two) Draw, approximately, the electric field lines about two signal charges, $+ Q$ and $- 3 Q ,$ which are a altitude $\ell$ apart.
Shital R.
Numerade Educator
Problem 30
(II) What is the electric field force at a indicate in space where a proton experiences an acceleration of ane.8 million "chiliad'due south"?
Ceren U.
Texas Tech University
Trouble 31
(1I) A long uniformly charged thread (linear charge density $\lambda = two.5 \mathrm { C } / \mathrm { yard } )$ lies along the $x$ axis in Fig. $56 .$ A small charged sphere $( Q = - 2.0 \mathrm { C } )$ is at the point $ten = 0 \mathrm { cm } , y = - 5.0 \mathrm { cm } .$ What is the electric field at the point $x = 7.0 \mathrm { cm } , \quad y = 7.0 \mathrm { cm } ?$ $\vec { \mathbf { Eastward } } _ { \text { thread } }$ and $\mathbf { \mathbf { Due east } } _ { \mathrm { O } }$ represent fields due to the long thread and the charge $Q ,$ respectively.
Eduard Southward.
Numerade Educator
Trouble 32
(Two) The electric field midway between two equal simply opposite betoken charges is $586 \mathrm { Northward } / \mathrm { C } ,$ and the distance between the charges is xvi.0$\mathrm { cm } .$ What is the magnitude of the charge on each?
Ceren U.
Texas Tech University
Trouble 33
(II) Summate the electric field at one corner of a square 1.22$\mathrm { thou }$ on a side if the other three corners are occupied by $ii.25 \times 10 ^ { - vi } \mathrm { C }$ charges.
Shital R.
Numerade Educator
Problem 34
(Ii) Calculate the electrical field at the center of a foursquare 52.5$\mathrm { cm }$ on a side if one corner is occupied by a $- 38.half-dozen \mu \mathrm { C }$ charge and the other iii are occupied past $- 27.0 \mu$ Charges.
Ceren U.
Texas Tech University
Trouble 35
(II) Determine the direction and magnitude of the electric field at the point $\mathrm { P }$ in Fig. $57 .$ The charges are separated past a altitude $two a ,$ and point $\mathrm { P }$ is a altitude $x$ from the midpoint betwixt the two charges. Limited your answer in terms of $Q , x , a ,$ and $k .$
Shital R.
Numerade Educator
Problem 36
(II) Two point charges, $Q _ { 1 } = - 25 \mu \mathrm { C }$ and $Q _ { two } = + 45 \mu \mathrm { C }$ are separated past a altitude of 12$\mathrm { cm } .$ The electric field at the point $\mathrm { P } ($ come across Fig. 58$)$ is zero. How far from $Q _ { 1 }$ is $\mathrm { P }$ ?
Ceren U.
Texas Tech University
Problem 37
(Ii) A very sparse line of charge lies along the $x$ axis from $ten = - \infty$ to $x = + \infty .$ Another similar line of accuse lies along the $y$ axis from $y = - \infty$ to $y = + \infty .$ Both lines have a uniform accuse per length $\lambda .$ Make up one's mind the resulting electric field magnitude and direction (relative to the $ten$ axis) at a indicate $( x , y )$ in the get-go quadrant of the $x y$ plane.
Shital R.
Numerade Educator
Problem 38
(2) $( a )$ Determine the electric field $\vec { \mathbf { East } }$ at the origin 0 in Fig. 59 due to the ii charges at $A$ and $B .$ (b) Repeat, only let the charge at B be reversed in sign.
Ceren U.
Texas Tech University
Problem 39
(II) Depict, approximately, the electrical field lines emanating from a uniformly charged straight wire whose length $\ell$ is not great. The spacing between lines well-nigh the wire should be much less than $\ell . [$Hint: Also consider points very far from the wire.]
Shital R.
Numerade Educator
Trouble 40
(Ii) Two parallel circular rings of radius $R$ have their centers on the $ten$ axis separated by a distance $\ell$ as shown in Fig. $60 .$ If each band carries a uniformly distributed accuse $Q ,$ find the electrical field, $\overline { \mathbf { Eastward } } ( x ) ,$ at indicate along the $x$ axis.
Ceren U.
Texas Tech Academy
Problem 41
(Two) You are given 2 unknown point charges, $Q _ { ane }$ and $Q _ { 2 }$ . At a point on the line joining them, one-3rd of the way from $Q _ { 1 }$ to $Q _ { 2 } ,$ the electrical field is zero (Fig. $61 ) .$ What is the ratio $Q _ { 1 } / Q _ { 2 } ?$
Shital R.
Numerade Educator
Problem 42
(Ii) Use Coulomb'due south law to determine the magnitude and management of the electric field at points $A$ and $B$ in Fig. 62 due to the two positive charges $( Q = 5.7 \mu C )$ shown. Are your results consequent with Fig. 34$b ?$
Ceren U.
Texas Tech Academy
Problem 43
(II) $( a )$ Two equal charges $Q$ are positioned at points $( x = \ell , y = 0 )$ and $( x = - \ell , y = 0 )$ . Determine the electric field as a role of $y$ for points along the $y$ centrality.(b) Testify that the field is a maximum at $y = \pm \ell / \sqrt { 2 }$ .
Pritesh R.
Numerade Educator
Problem 44
(II) At what position, $x = 10 _ { Chiliad } ,$ is the magnitude of the electric field forth the axis of the ring of Instance 9 of "Electric Accuse and Electrical Field" a maximum?
Ceren U.
Texas Tech University
Problem 45
(II) Guess the electric field at a point 2.xl$\mathrm { cm }$ perpendicular to the midpoint of a uniformly charged two.00 -thousand-long thin wire conveying a total charge of 4.75$\mu \mathrm { C }$ .
Shital R.
Numerade Educator
Problem 46
(2) The uniformly charged straight wire in Fig. 29 has the length $\ell ,$ where point 0 is at the midpoint. Show that the field at point $P$ , a perpendicular distance $x$ from $0 ,$ is given by $E = \frac { \lambda } { ii \pi \epsilon _ { 0 } } \frac { \ell } { ten \left( \ell ^ { 2 } + four x ^ { 2 } \right) ^ { one / 2 } }$ where $\lambda$ is the charge per unit length.
Ceren U.
Texas Tech University
Problem 47
(Two) Use your result from Problem 46 to find the electrical field (magnitude and direction) a altitude $z$ in a higher place the center of a square loop of wire, each of whose sides has length $\ell$ and uniform charge per length $\lambda$ (Fig. $63 ) .$
Shital R.
Numerade Educator
Problem 48
(2) Decide the direction and magnitude of the electrical field at the betoken $P$ shown in Fig. $64 .$ The 2 charges are separated by a distance of 2$a$ . Point $P$ is on the perpendicular bisector of the line joining the charges, a altitude $x$ from the midpoint between them. Limited your answers in terms of
$Q , ten , a ,$ and $yard .$
Ceren U.
Texas Tech Academy
Trouble 49
(III) A thin rod bent into the shape of an arc of a circle of radius $R$ carries a uniform accuse per unit length $\lambda .$ The arc subtends a total angle $2 \theta _ { 0 } ,$ symmetric about the $\frac { x } { x }$ axis, equally shown in Fig. $65 .$ Determine the electric field $\vec { \mathbf { E } }$ at the origin $0 .$
Linda W.
Numerade Educator
Problem 50
(3) A thin glass rod is a semicircle of radius $R ,$ Fig. $66 . \mathrm { A }$ charge is nonuniformly distributed along the rod with a linear charge density given by $\lambda = \lambda _ { 0 } \sin \theta ,$ where $\lambda _ { 0 }$ is a positive abiding. Point $P$ is at the eye of the semicircle. (a) Notice the electric field $\vec { \mathbf { Due east } }$ (magnitude and direction) at point $\mathrm { P } .$ [Hint: Remember $\sin ( - \theta ) = - \sin \theta ,$ so the two halves of the rod are oppositely charged.] (b) Determine the acceleration (magnitude and management) of an electron placed at betoken $\mathrm { P } ,$ assuming $R = i.0 \mathrm { cm }$ and $\lambda _ { 0 } = ane.0 \mu \mathrm { C } / \mathrm { chiliad }$
Ceren U.
Texas Tech Academy
Problem 51
(3) Suppose a uniformly charged wire starts at bespeak 0 and rises vertically along the positive $y$ axis to a length $\ell$ . (a) Determine the components of the electric field $Due east _ { ten }$ and $E _ { y }$ at bespeak $( x , 0 ) .$ That is, calculate $\vec { \mathbf { East } }$ near 1 finish of a long wire, in the plane perpendicular to the wire. (b) If the wire extends from $y = 0$ to $y = \infty ,$ so that $\ell = \infty ,$ show that $\vec { \mathbf { E } }$ makes a $45 ^ { \circ }$ bending to the horizontal for any $x .$ [Hint: See Example 11 of "Electrical Charge and Electric Field" andFig. $29 . ]$
Shital R.
Numerade Educator
Problem 52
(Iii) Suppose in Example 11 of "Electric Charge and Electrical Field" that $10 = 0.250 \mathrm { m } , Q = 3.15 \mu \mathrm { C } ,$ and that the uniformly charged wire is but six.fifty$\mathrm { m }$ long and extends forth the $y$ axis from $y = - 4.00 \mathrm { m }$ to $y = + ii.50 \mathrm { m }$ . (a) Calculate $East _ { ten }$ and $E _ { y }$ at point $\mathrm { P }$ (b) Determine what the mistake would be if yous just used the consequence of Instance $11 ,$ $Eastward = \lambda / ii \pi \epsilon _ { 0 } ten .$ Express this fault as $\left( Due east _ { x } - E \right) / Due east$ and
$E _ { y } / E .$
Ceren U.
Texas Tech University
Trouble 53
(III) A thin rod of length $\ell$ carries a full charge $Q$ distributed uniformly forth its length. Run into Fig. $67 .$ Determine the electric field along the axis of the rod starting at ane finish $-$ that is, observe $E ( x )$ for $10 \geq 0$ in Fig. 67
Linda W.
Numerade Educator
Problem 54
(III) Uniform plane of charge. Charge is distributed uniformly over a big square airplane of side $\ell$ , as shown in Fig. $68 .$ The charge per unit area $\left( \mathrm { C } / \mathrm { m } ^ { two } \right)$ is $\sigma .$ Decide the electric field at a point $\mathrm { P }$ a altitude $z$ higher up the center of the airplane, in the limit $\ell \rightarrow \infty .$ [Hint: Divide the plane into long narrow strips of width $d y ,$ and apply the result of Instance 11 of "Electric Accuse and Electric Field"; then sum the fields due to each strip to become the total field at P.]
Ceren U.
Texas Tech University
Problem 55
(III) Suppose the charge $Q$ on the ring of Fig. 28 was all distributed uniformly on only the upper one-half of the band, and no accuse was on the lower half. Determine the electric field $\vec { \mathbf { E } }$ at $\mathrm { P } .$ (Take $y$ vertically upward.)
Linda W.
Numerade Educator
Problem 56
(2) An electron with speed $v _ { 0 } = 27.v \times 10 ^ { 6 } \mathrm { m } / \mathrm { s }$ is traveling parallel to a compatible electric field of magnitude $E = 11.iv \times 10 ^ { 3 } \mathrm { North } / \mathrm { C }$ (a) How far will the electron travel before information technology stops? (b) How much time volition elapse before information technology returns to its starting point?
Ceren U.
Texas Tech Academy
Problem 57
(2) An electron has an initial velocity $\vec { \mathbf { v } } _ { 0 } = \left( 8.0 \times x ^ { 4 } \mathrm { m } / \mathrm { s } \right) \hat { \mathbf { j } }$ . It enters a region where $\quad \vec { \mathbf { E } } = ( ii.0 \lid { \mathbf { i } } + 8.0 \hat { \mathbf { j } } ) \times ten ^ { four } \mathrm { North } / \mathrm { C }$
(a) Decide the vector acceleration of the electron as a function of time. $( b )$ At what angle $\theta$ is it moving (relative to its initial direction) at $t = one.0 \mathrm { ns } ?$
Manish Yard.
Numerade Educator
Problem 58
(Ii) An electron moving to the right at $7.5 \times ten ^ { 5 } \mathrm { g } / \mathrm { southward }$ enters a uniform electric field parallel to its direction of motion. If the electron is to be brought to rest in the space of $four.0 \mathrm { cm } , ( a )$ what direction is required for the electric field, and $( b )$ what is the force of the field?
Ceren U.
Texas Tech Academy
Problem 59
(Two) At what angle will the electrons in Example xvi of "Electrical Charge and Electric Field" leave the uniform electric field at the end of the parallel plates (point $P$ in Fig. 41$) ?$ Presume the plates are 4.9$\mathrm { cm }$ long, $E = 5.0 \times x ^ { iii } \mathrm { Due north } / \mathrm { C } ,$ and $v _ { 0 } = 1.00 \times 10 ^ { 7 } \mathrm { thousand } / \mathrm { due south } .$ Ignore fringing of the field.
Aparna Southward.
Numerade Educator
Problem sixty
(II) An electron is traveling through a uniform electric field. The field is constant and given by $\overline { \mathbf { E } } =$ $\left( 2.00 \times ten ^ { - eleven } \mathrm { N } / \mathrm { C } \right) \chapeau { \mathbf { i } } - \left( one.twenty \times 10 ^ { - 11 } \mathrm { Northward } / \mathrm { C } \right) \lid { \mathbf { j } } .$ At $t = 0 ,$ the electron is at the origin and traveling in the $ten$ direction with a speed of 1.ninety$\mathrm { 1000 } / \mathrm { south }$ . What is its position 2.00 s later?
Ceren U.
Texas Tech Academy
Problem 61
(Ii) A positive charge $q$ is placed at the centre of a circular ring of radius $R$ . The ring carries a uniformly distributed negative accuse of total magnitude $- Q . ( a )$ If the charge $q$ is displaced from the center a small distance $ten$ as shown in Fig. $69 ,$ show that it will undergo uncomplicated harmonic motion when released. (b) If its mass is $m ,$ what is its menstruum?
Linda Due west.
Numerade Educator
Problem 62
(II) A dipole consists of charges $+ east$ and $- e$ separated by 0.68$\mathrm { nm } .$ It is in an electric field $E = ii.two \times 10 ^ { four } \mathrm { N } / \mathrm { C }$ . (a) What is the value of the dipole moment? (b) What is the torque on the dipole when it is perpendicular to the field? (c) What is the torque on the dipole when it is at an angle of $45 ^ { \circ }$ to the field? $( d )$ What is the work required to rotate the dipole from being oriented parallel to the field to beingness antiparallel to the field?
Ceren U.
Texas Tech University
Problem 63
(Ii) The HCl molecule has a dipole moment of about $three.4 \times 10 ^ { - thirty } \mathrm { C } \cdot \mathrm { m }$ . The two atoms are separated by near $1.0 \times 10 ^ { - x } \mathrm { m }$ (a) What is the cyberspace charge on each cantlet? (b) Is this equal to an integral multiple of $eastward ?$ If non, explain. (c) What maximum torque would this dipole feel in a $2.v \times ten ^ { four } \mathrm { N } / \mathrm { C }$ electric field? $( d )$ How much energy would be needed to rotate one molecule $45 ^ { \circ }$ from its equilibrium position of everyman potential free energy?
Stanley Eastward.
Numerade Educator
Problem 64
(2) Suppose both charges in Fig. 45 (for a dipole) were positive. (a) Prove that the field on the perpendicular bisector, for $r > \ell ,$ is given by $\quad \left( 1 / 4 \pi \epsilon _ { 0 } \right) \left( two Q / r ^ { 2 } \right) .$ (b) Explicate why the field decreases as 1$/ r ^ { 2 }$ here whereas for a dipole it decreases as 1$/ r ^ { iii }$ .
Ceren U.
Texas Tech University
Trouble 65
(II) An electrical dipole, of dipole moment $p$ and moment of inertia $I ,$ is placed in a uniform electrical field E. (a) If displaced by an angle $\theta$ equally shown in Fig. 44 and released, under what conditions will it oscillate in simple harmonic movement? (b) What will exist its frequency?
Shital R.
Numerade Educator
Problem 66
(III) Suppose a dipole $\vec { \mathbf { p } }$ is placed in a nonuniform electrical field $\vec { \mathbf { E } } = E \hat { \mathbf { i } }$ that points along the $x$ axis. If $\vec { \mathbf { E } }$ depends only on $\mathbf { 10 } ,$ show that the internet strength on the dipole is $\vec { \mathbf { F } } = \left( \vec { \mathbf { p } } \cdot \frac { d \vec { \mathbf { E } } } { d ten } \right) \hat { \mathbf { i } }$ where $d \vec { \mathbf { E } } / d 10$ is the gradient of the field in the $10$ direction.
Ceren U.
Texas Tech University
Problem 67
(III) (a) Show that at points forth the axis of a dipole (along the same line that contains $+ Q$ and $- Q ) ,$ the electric field has magnitude $East = \frac { i } { four \pi \epsilon _ { 0 } } \frac { 2 p } { r ^ { 3 } }$
Shital R.
Numerade Educator
Problem 68
How close must two electrons be if the electrical force between them is equal to the weight of either at the Earth'south surface?
Ceren U.
Texas Tech University
Problem 69
Given that the human body is by and large made of water, judge the full corporeality of positive charge in a 65 -kg person.
Shital R.
Numerade Educator
Problem 70
A iii.0 -g copper penny has a positive charge of 38$\mu \mathrm { C }$ . What fraction of its electrons has it lost?
Ceren U.
Texas Tech Academy
Problem 71
Measurements indicate that there is an electric field surrounding the Earth. Its magnitude is most 150$\mathrm { Northward } / \mathrm { C }$ at the Earth's surface and points inward toward the Earth'southward heart. What is the magnitude of the electric charge on the Earth? Is it positive or negative? [Hint: The electric field outside a uniformly charged sphere is the same as if all the accuse were concentrated at its centre.
Shital R.
Numerade Educator
Problem 72
(a) The electrical field near the Earth's surface has magnitude of nigh 150$\mathrm { N } / \mathrm { C }$ . What is the acceleration experienced by an electron near the surface of the Earth? $( b )$ What almost a proton? $( c )$ Calculate the ratio of each acceleration to $g = 9.viii \mathrm { 1000 } / \mathrm { due south } ^ { 2 }$
Ceren U.
Texas Tech Academy
Trouble 73
A water droplet of radius 0.018$\mathrm { mm }$ remains stationary in the air. If the downward-directed electric field of the Earth is $150 \mathrm { Due north } / \mathrm { C } ,$ how many excess electron charges must the water droplet accept?
Shital R.
Numerade Educator
Problem 74
Estimate the net force betwixt the $\mathrm { CO }$ group and the HN group shown in Fig. $70 .$ The $\mathrm { C }$ and $\mathrm { O }$ have charges $\pm 0.twoscore e ,$ and the $\mathrm { H }$ and $\mathrm { N }$ have charges $\pm 0.xx \mathrm { e } ,$ where $eastward = ane.6 \times 10 ^ { - nineteen } \mathrm { C } . [$Hint: Do not include the "internal" forces between $\mathrm { C }$ and $\mathrm { O } ,$ or between $\mathrm { H }$ and $\mathrm { North } . ]$
Ceren U.
Texas Tech University
Problem 75
Suppose that electrical attraction, rather than gravity, were responsible for holding the Moon in orbit around the Earth. If equal and opposite charges $Q$ were placed on the Globe and the Moon, what should be the value of $Q$ to maintain the present orbit? Treat the World and Moon every bit point particles.
Shital R.
Numerade Educator
Problem 76
In a uncomplicated model of the hydrogen atom, the electron revolves in a round orbit around the proton with a speed of $2.2 \times 10 ^ { 6 } \mathrm { thousand } / \mathrm { s }$ . Determine the radius of the electron's orbit. [ Hint: Call back circular motion.
Ceren U.
Texas Tech University
Problem 77
A positive signal accuse $Q _ { 1 } = two.5 \times 10 ^ { - 3 } \mathrm { C }$ is fixed at the origin of coordinates, and a negative bespeak charge $Q _ { 2 } = - 5.0 \times 10 ^ { - half dozen } \mathrm { C }$ is fixed to the $x$ axis at $x = + 2.0 \mathrm { grand }$ . Find the location of the identify(s) along the $10$ axis where the electrical field due to these two charges is zippo.
Shital R.
Numerade Educator
Trouble 78
When dress are removed from a dryer, a $40 - \mathrm { m }$ sock is stuck to a sweater, even with the sock clinging to the sweater's underside. Estimate the minimum attractive force betwixt the sock and the sweater. And so estimate the minimum charge on the sock and the sweater. Presume the charging came entirely from the sock rubbing against the sweater and so that they have equal and opposite charges, and estimate the sweater equally a flat sheet of uniform charge.
Ceren U.
Texas Tech University
Problem 79
A small lead sphere is encased in insulating plastic and suspended vertically from an platonic leap (spring abiding $k = 126 \mathrm { N } / \mathrm { m } )$ as in Fig. $71 .$ The total mass of the coated sphere is $0.650 \mathrm { kg } ,$ and its eye lies 15.0$\mathrm { cm }$ above a tabletop when in equilibrium. The sphere is pulled downwardly 5.00$\mathrm { cm }$ below equilibrium, an electrical charge $Q = - three.00 \times 10 ^ { - 6 } \mathrm { C }$ is deposited on it, and then it is released. Using what y'all know about harmonic oscillation, write an expression for the electric field forcefulness as a function of time that would exist measured at the point on the tabletop (P) directly below the sphere.
Shital R.
Numerade Educator
Problem 80
A large electroscope is made with "leaves" that are 78 -cm-long wires with tiny 24 g spheres at the ends. When charged, nearly all the accuse resides on the spheres. If the wires each brand a $26 ^ { \circ }$ angle with the vertical (Fig. $72 ) ,$ what total accuse $Q$ must have been applied to the electroscope? Ignore the mass of the wires.
Ceren U.
Texas Tech Academy
Problem 81
Dry air volition break down and generate a spark if the electric field exceeds almost $3 \times x ^ { 6 } \mathrm { N } / \mathrm { C } .$ How much charge could
be packed onto a dark-green pea (diameter 0.75$\mathrm { cm }$ ) before the pea spontaneously discharges? [Hint: Eqs. four work outside a sphere if $r$ is measured from its center.] $E = \frac { F } { q } = \frac { k q Q / r ^ { 2 } } { q }$
$Eastward = k \frac { Q } { r ^ { 2 } }$ or, in terms of $\epsilon _ { 0 }$ every bit in Eq. $2 \left( k = 1 / 4 \pi \epsilon _ { 0 } \right) :$ $Due east = \frac { 1 } { 4 \pi \epsilon _ { 0 } } \frac { Q } { r ^ { 2 } }$
Shital R.
Numerade Educator
Problem 82
2 point charges, $Q _ { i } = - vi.7 \mu \mathrm { C }$ and $Q _ { 2 } = ane.8 \mu \mathrm { C } ,$ are located between two oppositely charged parallel plates, equally shown in Fig. $73 .$ The two charges are separated past a distance of $10 = 0.34 \mathrm { grand } .$ Assume that the electric field produced by the charged plates is uniform and equal to $E = 73,000 \mathrm { N } / \mathrm { C } .$ Calculate the net electrostatic force on $Q _ { 1 }$ and give its direction.
Ceren U.
Texas Tech University
Trouble 83
Packing material made of pieces of foamed polystyrene can hands become charged and stick to each other. Given that the density of this textile is about 35$\mathrm { kg } / \mathrm { grand } ^ { three }$ , gauge how much charge might be on a 2.0 -cm-diameter foamed polystyrene sphere, bold the electrical force between two spheres stuck together is equal to the weight of one sphere.
Shital R.
Numerade Educator
Problem 84
One type of electric quadrupole consists of ii dipoles placed finish to end with their negative charges (say) overlapping; that is, in the middle is $- two Q$ flanked (on a line) by a $+ Q$ to either side (Fig. $74 ) .$ Determine the electrical field $\vec { \mathbf { East } }$ at points along the perpendicular bisector and show that $\mathrm { East } decreases as 1$/ r ^ { four } .$ Measure $r$ from the $- 2 Q$ charge and assume $r \gg \ell$
Ceren U.
Texas Tech University
Trouble 85
Suppose electrons enter a uniform electric field midway between two plates at an bending $\theta _ { 0 }$ to the horizontal, equally shown in Fig. $75 .$ The path is symmetrical, then they leave at the same angle $\theta _ { 0 }$ and simply barely miss the top plate. What is $\theta _ { 0 } ?$ Ignore fringing of the field.
Shital R.
Numerade Educator
Problem 86
An electron moves in a circle of radius $r$ effectually a very long uniformly charged wire in a vacuum bedchamber, as shown in Fig. $76 .$ The charge density on the wire is $\lambda = 0.14 \mu \mathrm { C } / \mathrm { m } .$ (a) What is the electric field at the electron (magnitude and management in terms of $r$ and $\lambda ) ? ( b )$ What is the speed of the electron?
Ceren U.
Texas Tech University
Trouble 87
Iii very large foursquare planes of charge are arranged as shown (on edge) in Fig. $77 .$ From left to right, the planes have charge densities per unit area of $- 0.50 \mu C / m ^ { 2 }$ $+ 0.25 \mu C / yard ^ { 2 } ,$ and $- 0.35 \mu C / m ^ { 2 } .$ Find the total electric field (direction and magnitude) at the points $A , B , C ,$ and $D$ . Presume the plates are much larger than the distance AD.
Shital R.
Numerade Educator
Trouble 88
A bespeak charge $( 1000 = ane.0 \mathrm { one thousand } )$ at the end of an insulating cord of length 55$\mathrm { cm }$ is observed to be in equilibrium in a uniform horizontal electric field of $fifteen,000 \mathrm { N } / \mathrm { C }$ , when the pendulum's position is equally shown in Fig. $78 ,$ with the charge 12$\mathrm { cm }$ above the lowest (vertical) position. If the field points to the right in Fig. 78 , make up one's mind the magnitude and sign of the point accuse.
Ceren U.
Texas Tech University
Problem 89
Iv equal positive point charges, each of accuse $8.0 \mu \mathrm { C } ,$ are at the corners of a square of side 9.2$\mathrm { cm } .$ What accuse should be placed at the center of the square so that all charges are at equilibrium? Is this a stable or unstable equilibrium in the plane?
Shital R.
Numerade Educator
Trouble 90
Two pocket-size, identical conducting spheres $A$ and $B$ are a distance $R$ apart; each carries the aforementioned charge $Q . ( a )$ What is the force sphere B exerts on sphere A? (b) An identical sphere with zero charge, sphere $C$ , makes contact with sphere $B$ and is then moved very far abroad. What is the net strength now interim on sphere A? (c) Sphere $C$ is brought back and now makes contact with sphere $A$ and is then moved far away. What is the forcefulness on sphere $A$ in this third case?
Ceren U.
Texas Tech University
Problem 91
A point accuse of mass 0.210$\mathrm { kg }$ , and net charge $+ 0.340 \mu \mathrm { C }$ , hangs at rest at the end of an insulating cord higher up a large sheet of charge. The horizontal sheet of fixed uniform accuse creates a uniform vertical electric field in the vicinity of the point charge. The tension in the string is measured to exist 5.eighteen$\mathrm { N }$ . (a) Calculate the magnitude and direction of the electrical field due to the sail of charge (Fig. 79). (b) What is the surface charge density $\sigma \left( \mathrm { C } / \mathrm { m } ^ { 2 } \right)$ on the sheet?
Shital R.
Numerade Educator
Problem 92
A i-dimensional row of positive ions, each with accuse $+ Q$ and separated from its neighbors by a distance $d$ , occupies the correct-hand half of the $x$ axis. That is, at that place is a $+ Q$ charge at $10 = 0 , 10 = + d , x = + 2 d , x = + iii d$ and so $+ Q$ accuse at $x = 0 , x = + d , ten = + 2 d , 10 = + 3 d$ and so on out to $\infty . ( a )$ If an electron is placed at the position $10 = - d ,$ make up one's mind $F ,$ the magnitude of force that this row of charges exerts on the electron. (b) If the electron is instead placed at $x = - 3 d ,$ what is the value of $F ?$ [Hint: The infinite sum $\sum _ { northward = one } ^ { due north = \infty } \frac { i } { n ^ { ii } } = \frac { \pi ^ { ii } } { 6 } ,$ where $n$ is a positive integer. $]$
Ceren U.
Texas Tech University
Trouble 93
(III) A sparse ring-shaped object of radius $a$ contains a full charge $Q$ uniformly distributed over its length. The electrical field at a signal on its axis a distance $ten$ from its center is given in Instance 9 of "Electrical Charge and Electric Field" equally $East = \frac { 1 } { 4 \pi \epsilon _ { 0 } } \frac { Q x } { \left( x ^ { 2 } + a ^ { 2 } \correct) ^ { \frac { 3 } { two } } }$
(a) Take the derivative to notice where on the $x$ axis $( ten > 0 ) E _ { x }$ is a maximum. Presume $Q = 6.00 \mu \mathrm { C }$ and $a = 10.0 \mathrm { cm } .$
(b) Calculate the electrical field for $ten = 0$ to $ten = + 12.0 \mathrm { cmin }$ steps of $0.one \mathrm { cm } ,$ and make a graph of the electric field. Does the maximum of the graph coincide with the maximum of the electric field you obtained analytically? As well, summate and graph the electric field $( c )$ due to the ring, and $( d )$ due to a point charge $Q = six.00 \mu C$ at the center of the ring. Make a single graph, from $x = 0$ (or $x = ane.0 \mathrm { cm }$ ) out to $10 = 50.0 \mathrm { cm }$ in one.0$\mathrm { cm }$ steps, with 2 curves of the electric fields, and show that both fields converge at big distances from the center. (due east) At what distance does the electric field of the ring differ from that of the bespeak accuse by 10$\% ?$
Shital R.
Numerade Educator
Problem 94
(3) $\mathrm { An } viii.00 \mu \mathrm { C }$ charge is on the $x$ axis of a coordinate system at $x = + 5.00 \mathrm { cm } . \mathrm { A } - two.00 \mu \mathrm { C }$ charge is at $x = - five.00 \mathrm { cm }$ (a) Plot the $ten$ component of the electric field for points on the $x$ axis from $x = - 30.0 \mathrm { cm }$ to $x = + thirty.0 \mathrm { cm } .$ The sign of $Eastward _ { x }$ is positive when $\vec { \mathbf { East } }$ points to the right and negative when it points to the left. $( b )$ Brand a plot of $E _ { 10 }$ and $E _ { y }$ for
points on the $y$ axis from $y = - thirty.0$ to $+ 30.0 \mathrm { cm } .$
Ceren U.
Texas Tech University
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Source: https://www.numerade.com/books/chapter/electric-charge-and-electric-field-4/