Essential University Physics, 3e (Wolfson) 3e (Wolfson) Chapter 22 Electric Potential
2 2. 1 Co nc e pt ua lQue s t i o ns 1) If the electric field is zero everywhere inside a region of space, the potential must also be zero in that region. A) True B) alse Answer! B "ar! 1
#) $hen the electric field is zero at a point, the potential must also be zero there. A) True B) alse Answer! B "ar! 1
%) If the electrical potential in a region is constant, the electric field must be zero everywhere in that region. A) True B) alse Answer! A "ar! 1
&) If the electric potential at a point po int in space is zero, then the electric field at that point must also be zero. A) True B) alse Answer! B "ar! 1
') A negative charge, if free, will tend to move A) from high potential to low potential. B) from low potential to high potential. () toward infinity. infinity. ) away from infinity. *) in the direction of the electric field. Answer! B "ar! 1
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-) /uppose a region of space has a uniform electric field, directed towards the right, as shown in the figure. $hich statement about the electric potential is true0
A) The potential at all three locations A, un iform. A, B, B, C ) is the same because the field is uniform. B) The potential at points A points A and and B B are are e2ual, and the potential at point C is is higher than the potential at point A point A.. () The potential at points A points A and and B B are are e2ual, and the potential at point C is is lower than the potential at point A point A.. ) The potential at point A point A is is the highest, the potential at point B point B is is the second highest, and the potential at point C is is the lowest. Answer! ( "ar! 1
3) $hich statements are true for an electron moving in the direction of an electric field0 There may be more than one correct choice.) A) Its electric potential energy increases as it goes from high to low potential. B) Its electric potential energy decreases as it goes from high to low potential. () Its potential energy increases as its 4inetic energy decreases. ) Its 4inetic energy decreases as it moves in the direction of the electric field. *) Its 4inetic energy increases as it moves in the direction of the electric field. Answer! A, (, "ar! 1
5) /uppose you have two point charges of opposite sign. As you move them farther and farther apart, the potential energy of this system relative to infinity A) increases. B) decreases. () stays the same. Answer! A "ar! 1
6) /uppose you have two negative point charges. As you move them farther and farther apart, the potential energy of this system system relative to infinity A) increases. B) decreases. () stays the same. Answer! B "ar! 1
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1) Two e2ual positive charges are held in place at a fi7ed distance. If you put a third positive charge midway between these two charges, its electrical potential energy of the system relative to infinity) is zero because the electrical forces on the third charge due to the two fi7ed charges 8ust balance each other. A) True B) alse Answer! B "ar! 1
11) A metallic sphere of radius ' cm is charged such that the potential of its surface is 1 " relative to infinity). $hich of the following plots correctly shows the potential as a function of distance from the center of the sphere0
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A) plot $ B) plot 9 () plot : ) plot ; Answer! B "ar! 1
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1#) A conducting sphere of radius R carries an e7cess positive charge and is very far from any other charges. $hich one of the following graphs best illustrates the potential relative to infinity) produced by this sphere as a function of the distance r from the center of the sphere0 A)
B)
()
)
*)
Answer! "ar! 1
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1%) A nonconducting sphere contains positive charge distributed uniformly throughout its volume. $hich statements about the potential due to this sphere are true0 All potentials are measured relative to infinity. There may be more than one correct choice.) A) The potential is highest at the center of the sphere. B) The potential at the center of the sphere is zero. () The potential at the center of the sphere is the same as the potential at the surface. ) The potential at the surface is higher than the potential at the center. *) The potential at the center is the same as the potential at infinity. Answer! A "ar! 1
1&) A conducting sphere contains positive charge distributed uniformly over its surface. $hich statements about the potential due to this sphere are true0 All potentials are measured relative to infinity. There may be more than one correct choice.) A) The potential is lowest, but not zero, at the center of the sphere. B) The potential at the center of the sphere is zero. () The potential at the center of the sphere is the same as the potential at the surface. ) The potential at the surface is higher than the potential at the center. *) The potential at the center is the same as the potential at infinity. Answer! ( "ar! 1
1') A negative charge is moved from point A to point B along an e2uipotential surface. $hich of the following statements must be true for this case0 A) The negative charge performs wor4 in moving from point A to point B. B) $or4 is re2uired to move the negative charge from point A to point B. ()
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1-) The graph in the figure shows the variation of the electric potential V measured in volts) as a function of the radial direction r measured in meters). or which range or value of r is the magnitude of the electric field the largest0
A) from r = m to r = % m B) from r = % m to r = & m () from r = & m to r = - m ) at r = % m *) at r = & m Answer! B "ar! 1
13) The graph in the figure shows the variation of the electric potential V x) in arbitrary units) as a function of the position x also in arbitrary units). $hich of the choices below correctly describes the orientation of the x-component of the electric field along the x-a7is0
A) E x is positive from x = ># to x = #. B) E x is positive from x = ># to x = , and negative from x = to x = #. () E x is negative from x = ># to x = , and positive from x = to x = #. ) E x is negative from x = ># to x = #. Answer! B "ar! 1
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15) The potential as a function of position x is shown in the graph in the figure. $hich statement about the electric field is true0
A) The electric field is zero at x = , its magnitude is at a ma7imum at x = ' cm, and the field is directed to the right there. B) The electric field is zero at x = ' cm, its magnitude is at a ma7imum at x = , and the field is directed to the right there. () The electric field is zero at x = , its magnitude is at a ma7imum at x = 1' cm, and the field is directed to the left there. ) The electric field is zero at x = 1 cm, its magnitude is at a ma7imum at x = ' cm, and the field is directed to the left there. Answer! A "ar! 1
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2 2 . 2 Pr o bl e ms 1) Two positive point charges ?&. @( and ?#. @( are placed at the opposite corners of a rectangle as shown in the figure. k = 1&πε = 5.66 16 < C m#(#)
a) $hat is the potential at point A relative to infinity) due to these charges0 b) $hat is the potential at point B relative to infinity) due to these charges0 Answer! a) ?5.66 1& " b) 1.1# 1' " "ar! 1
#) Three point charges of >#. @(, ?&. @(, a nd ?-. @( are placed along the x>a7is as shown in the figure. $hat is the electrical potential at point P relative to infinity) due to these charges0 k = 1&πε = 5.66 16 < C m#(#)
A) >%3 4" B) ?%3 4" () >1'& 4" ) ?1'& 4" *) 4" Answer! B "ar! 1
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%) our e2ual ?-.>@( point charges are placed at the corners of a s2uare #. m on each side. k = 1&πε = 5.66 16 < C m#(#) a) $hat is the electric potential relative to infinity) due to these charges at the center of this s2uare0 b) $hat is the magnitude of the electric field due to these charges at the center of the s2uare0 Answer! a) 1'% 4" b) zero "ar! 1
&) our point charges of magnitude -. @( and of varying signs are placed at the corners of a s2uare #. m on each side, as shown in the figure. k = 1&πε = 5.66 16 < C m#(#) a) $hat is the electric potential relative to infinity) at the center of this s2uare due to these charges0 b) $hat is the magnitude of the electric field due to these charges at the center of the s2uare0
Answer! a) zero
b) 3.-% 1& <(
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') Two ?-.>@( point charges are placed at the corners of the base of an e2uilateral triangle, as shown in the figure. k = 1&πε = 5.66 16 < C m#(#) At the verte7, P , of the triangle
a) what is the electric potential relative to infinity) due to these charges0 b) what is the magnitude of the electric field due to these charges0 Answer! a) '& 4" b) #.% 1& <( "ar! 1
-) A ?&. @(>point charge and a >&.>@( point charge are placed as shown in the figure. $hat is the potential difference, V A > V B, between points A and B0 k = 1&πε = 5.66 16 < C m#(#)
A) &5 " B) 6- " () . " ) 6- 4" *) &5 4" Answer! "ar! 1
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3) Two point charges of ?#. @( and >-. @( are located on the x>a7is at x = >1. cm and x = ?#. cm respectively. $here should a third charge of ?%.>@( be placed on the ? x>a7is so that the potential at the origin is e2ual to zero0 k = 1&πε = 5.66 16 < C m#(#) A) x = &. cm B) x = 1. cm () x = #. cm ) x = %. cm *) x = '. cm Answer! "ar! 1
5) A >3.>@( point charge has a positively charged ob8ect in an elliptical orbit around it. If the mass of the positively charged ob8ect is 1. 4g and the distance varies from '. mm to #. mm between the charges, what is the ma7imum electric potential difference through which the positive ob8ect moves0 k = 1&πε = 5.66 16 < C m#(#) A) 6.& D" B) %.# D" () &.# D" ) 1- D" Answer! A "ar! 1
6) The figure shows two arcs of a circle on which charges ?Q and >Q have been spread uniformly. $hat is the value of the electric potential at the center of the circle0
Answer! ;ero "ar! 1
1) A half>ring semicircle) of uniformly distributed charge Q has radius R. $hat is the electric potential at its center0 Answer! E&π ε R "ar! 1
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11) A very small ob8ect carrying >-. @( of charge is attracted to a large, well>anchored, positively charged ob8ect. Fow much 4inetic energy does the negatively charged ob8ect gain if the potential difference through which it moves is %. m"0 k = 1&πε = 5.66 16 < C m#(#) A) 15 nG B) .' 4G () .' G ) -. @G Answer! A "ar! %
1#) Two point charges of ?1. @( and >#. @( are located .' m apart. $hat is the minimum amount of wor4 needed to move the charges apart to double the distance between them0 k = 1&πε = 5.66 16 < C m#(#) A) >%- mG B) ?15 mG () mG ) ?%- mG *) >15 mG Answer! B "ar! 1
1%) Two e2ual point charges Q are separated by a distance d . Hne of the charges is released and moves away from the other due only to the electrical force between them. $hen the moving charge is a distance %d from the other charge, what is its 4inetic energy0 Answer! Q#- Jd "ar! 1
1&) If an electron is accelerated from rest through a potential difference of 6.6 4", what is its resulting speed0 e = 1.- 1>16 (, k = 1&πε = 5.66 16 < C m#(#, mel = 6.11 7 1>%1 4g) A) '.6 13 ms B) &.6 13 ms () %.6 13 ms ) #.6 13 ms Answer! A "ar! 1
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1') (onsider the group of three?#.& n( point charges shown in the figure. $hat is the electric potential energy of this system of charges relative to infinity0 k = 1&πε = 5.66 16 < C m#(#)
A) &.1 1>- G B) &.- 1>- G () &.# 1>- G ) &.& 1>- G Answer! A "ar! '?
1-) An electron is released from rest at a distance of 6. cm from a proton. If the proton is held in place, how fast will the electron be moving when it is %. cm from the proton0 mel = 6.11 7 1>%1 4g, e = 1.- 1>16 (, k = 1&πε = 5.66 16 < C m#(#) A) 3'. ms B) 1- ms () 1% ms ) 1.- × 1% ms *) &.-& × 1' ms Answer! B "ar! 1
13) A >%.>@( point charge and a >6.>@( point charge are initially e7tremely far apart. Fow much wor4 does it ta4e to bring the >%.>@( charge to x = %. mm, y = . mm and the >6.>@( charge to x = >%. mm, y = . mm0 k = 1&πε = 5.66 16 < C m#(#) A) & G B) 51 G () #3 G ) -.5 G Answer! A "ar! 1
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15) A tiny ob8ect carrying a charge of ?%. @( and a second tiny charged ob8ect are initially very far apart. If it ta4es of wor4 to bring them to a final configuration in which the ob8ect i is at x = 1. mm, y = 1. mm, and the other charged ob8ect is at x = 1. mm, y = %. mm, find the magnitude of the charge on the second ob8ect. k = 1&πε = 5.66 16 < C m#(#) A) #.1' @( B) &.% @( () 1.3& @( ) &.% n( Answer! A "ar! '?
16) The figure shows an arrangement of two >&.' n( charges, each separated by '. mm from a proton. If the two negative charges are held fi7ed at their locations and the proton is given an initial velocity v as shown in the figure, what is the minimum initial speed v that the proton needs to totally escape from the negative charges0 k = 1&πε = 5.66 16 < C m#(#, e = 1.- 1> 16 (, m proton = 1.-3 7 1>#3 4g)
A) 1.5 1- ms B) %.' 1- ms () -.5 1- ms ) 1.& 13 ms Answer! A "ar! '?
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#) An alpha particle is a nucleus of helium. It has twice the charge and four times the mass of the proton. $hen they were very far away from each other, but headed toward directly each other, a proton and an alpha particle each had an initial speed of .%c, where c is the speed of light. $hat is their distance of closest approach0 Hint: There are two conserved 2uantities. Da4e use of both of them. c = %. 15 ms, k = 1&πε = 5.66 16 < C m#(#, e = 1.- 1>16 (, m proton = 1.-3 7 1>#3 4g) A) #.1 1>1% m B) %.% 1>1% m () #.- 1>1% m ) #.6 1>1% m Answer! A "ar! '?
#1) Two point charges, Q and >%Q, are located on the x>a7is a distance d apart, with >%Q to the right of Q. ind the location of AKK the points on the x>a7is not counting infinity) at which the potential relative to infinity) due to this pair of charges is e2ual to zero. Answer! d & to the right of Q between the charges) and d # to the left of Q "ar! 1
##) A sphere with radius #. mm carries ?1. @( of charge distributed uniformly throughout its volume. $hat is the potential difference, V B - V A, between point B, which is &. m from the center of the sphere, and point A, which is 6. m from the center of the sphere0 k = 1&πε = 5.66 16 < C m#(#) A) 1# " B) >1# " () 1& " ) >.&' " Answer! A "ar! 1
#%) A conducting sphere is charged up such that the potential on its surface is 1 " relative to infinity). If the sphereLs radius were twice as large, but the charge on the sphere were the same, what would be the potential on the surface relative to infinity0 A) ' " B) #' " () 1 " ) # " Answer! A "ar! 1
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#&) A conducting sphere of radius #. cm carries an e7cess charge of ?1'. M(, and no other charges are present. k = 1&πε = 5.66 16 < C m#(#) The potential relative to infinity) due to this sphere at a point 1#. cm from its center is closest to A) zero. B) -3& 4". () 11% 4". ) %%5 4". *) 6%5 4". Answer! B "ar! 1
#') A conducting sphere &' cm in diameter carries an e7cess of charge, and no other charges are present. :ou measure the potential of the surface of this sphere and find it to be 1& 4" relative to infinity. k = 1&πε = 5.66 16 < C m#(#) The e7cess charge on this sphere is closest to A) .%' n(. B) 36 n(. () %1' n(. ) %' n(. *) 3 n(. Answer! "ar! 1
#-) An e7tremely long thin wire carries a uniform linear charge density of %'5 n(m. ind the potential difference between points '. m and -. m from the wire, provided they are not near either end of the wire. k = 1&πε = 5.66 16 < C m#(#) A) 1.# 4" B) -. 4" () .#1 4" ) .#1' 4" Answer! A "ar! 1
#3) Two long conducting cylindrical shells are coa7 ial and have radii of # mm and 5 mm. The electric potential of the inner conductor, with respect to the outer conductor, is ?- ". An electron is released from rest at the surface of the ou ter conductor. $hat is the speed of the electron as it reaches the inner conductor0 e = 1.- 1>16 (, mel = 6.11 7 1>%1 4g, k = 1&πε = 5.66 16 < C m#(#) A) 1.1 × 13 ms B) 1.% × 13 ms () 1.' × 13 ms ) 1.3 × 13 ms *) 1.6 × 13 ms Answer! ( "ar! 1
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#5) Two long conducting cylindrical shells are coa7 ial and have radii of # mm and 5 mm. The electric potential of the inner conductor, with respect to the outer conductor, is ?- ". $hat is the ma7imum electric field magnitude between the cylinders0 k = 1&πε = 5.66 16 < C m#(#) A) 1, "m B) 1&, "m () 15, "m ) ##, "m *) #-, "m Answer! "ar! 1
#6) A very long nonconducting cylinder of diameter 1. cm carries charge distributed uniformly over its surface. *ach meter of length carries ?'.' M( of charge. A proton is released from rest 8ust outside the surface. Fow far will it be from the /NOA(* of the cylinder when its speed has reached #'' 4ms0 k = 1&πε = 5.66 16 < C m#(#, e = 1.- 1>16 (, m proton = 1.-3 7 1>#3 4g) Answer! #.' cm "ar! 1
%) Two parallel conducting plates are separated by and carry e2ual but opposite surface charge densities. If the potential difference between them is what is the magnitude of the surface charge density on each plate0 ε = 5.5' 1>1# (#< C m#) A) 15 n(m# B) .1% m(m# () %' n(m# ) .#3 m(m# Answer! A "ar! '
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%1) Two large conducting parallel plates A and B are separated by #.& m. A uniform field of 1' "m, in the positive x>direction, is produced by charges on the plates. The center plane at x = . m is an e2uipotential surface on which V = . An electron is pro8ected from x = . m, with an initial velocity of 1. 13 ms perpendicular to the plates in the positive x>direction, as shown in the figure. $hat is the 4inetic energy of the electron as it reaches plate A0 e = 1.- 1>16 (, mel = 6.11 7 1>%1 4g)
A) ?#.& × 1>1- G B) ?%.% × 1>1- G () >#.& × 1>1- G ) >#.6 × 1>1- G *) >%.% × 1>1- G Answer! B "ar! 1
%#) A charge Q = >5# n( is uniformly distributed on a ring of #.& m radius. A point charge q = ?'% n( is fi7ed at the center of the ring. oints A and B are located on the a7is of the ring, as shown in the figure. $hat is the minimum wor4 that an e7ternal force must do to transport an electron from B to A0 e = 1.- 1>16 (, k = 1&πε = 5.66 16 < C m#(#)
A) >5.3 1>13 G B) ?3.# 1>15 G () ?1. 1>1- G ) ?5.3 1>13 G *) >3.# 1>15 G Answer! A "ar! 1 16 (opyright + #1- earson *ducation, Inc.
%%) A charge Q = >-1 n( is uniformly distributed on a ring of #.&>m radius. A point charge q = ?&5 n( is fi7ed at the center of the ring, as shown in the figure. An electron is pro8ected from infinity toward the ring along the a7is of the ring. This electron comes to a momentary halt at a point on the a7is that is '. m from the center of the ring. $hat is the initial speed of the electron at infinity0 e = 1.- 1>16 (, k = 1&πε = 5.66 16 < C m#(#, mel = 6.11 7 1>%1 4g)
A) -.- 1- ms B) &.' 1- ms () %.& 1- ms ) #.# 1- ms *) 1.1 1- ms Answer! A "ar! 1
%&) If the electric potential in a region is given by V x) = - x#, the x component of the electric field in that region is A) >1# x>%. B) >- x. () 1# x>%. ) 1# x. *) - x. Answer! ( "ar! 1
%') If the potential in a region is given by V x, y, z ) = xy > % z >#, then the y component of the electric field in that region is A) x y - - z >%. B) -y. () > x. ) x y. Answer! ( "ar! 1
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%-) In a certain region, the electric potential due to a charge distribution is given b y the e2uation V x!y!z ) = % x# y# ? yz % > # z % x, where x! y! and z are measured in meters and V is in volts. (alculate the magnitude of the electric field vec tor at the position x!y!z ) = 1., 1., 1.). A) &.% "m B) #. "m () >5.1 "m ) 5.- "m *) 3& "m Answer! "ar! 1
%3) In a certain region, the electric potential due to a charge distribution is given b y the e2uation V x!y) = # xy > x# > y, where x and y are measured in meters and V is in volts. At which point is the electric field e2ual to zero0 A) x = .' m, y = 1 m B) x = 1 m, y = 1 m () x = 1 m, y = .' m ) x = .' m, y = .' m *) x = m, y = m Answer! "ar! 1
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