Download Exam I, vers. 0002 - Physics 1120

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Exam I, vers. 0002 - Physics 1120 - Fall, 2007
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1. Two boxes of unequal masses M1 and M2 have been placed on a flat frictionless surface some
distance away from each other, and given unequal positive charges Q1 and Q2. The first box
is heavier than the second, M1 > M2, however the charge on the first box is smaller, Q1 < Q2.
Which of the following statements are true?
M1, Q1
M2, Q2
By Newton’s third law, the force acting on each box is the same regardless of its charge.
Thus the lighter box has bigger acceleration.
A) Both boxes start accelerating, and the first one (M1) has bigger acceleration.
B) Both boxes start accelerating, and the second one (M2) has bigger acceleration.
C) The accelerations are equal and nonzero.
D) Neither of the boxes accelerate.
E) None of the above
2. A balloon of mass M has been positively charged (so that its charge is Q) and placed next to a
vertical blackboard, as shown. The blackboard polarizes, so the balloon sticks to it. The
friction coefficient between the balloon and the blackboard is µ, and we will assume that the
electric field created by the blackboard is uniform and directed towards the blackboard
perpendicular to it. What is the magnitude E of this electric field? The electric force presses
the balloon against the board with the force EQ. This creates the friction force µEQ
which balances the gravitational force Mg. Thus Mg=µEQ.
A)
B)
C)
µMg
Q
Q
E=
µMg
µQ
E=
Mg
E=
D)
E=
Mg
µQ
E)
µM
E=
gQ
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3.Two positive +Q charges and one negative -2Q
charge are placed as shown on the picture. An
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electron of negative charge -e is placed as shown. Where is the force acting on the electron
directed?
y
A) to the right (positive x direction)
B) upwards (positive y direction)
C) to the left (negative x direction)
D) downwards (negative y
direction)
E) the force is zero
+Q
−2Q
30◦
r
The electron gets repelled from the -2Q charge,
but is attracted to the two +Q charges. However,
those two are farther away, the sum of the forces from
the two +Q charges is not enough to counteract the force
charge.
x
30◦
+Q
since
from the -2Q
electron
4. A small positive test charge +q flies by a large positive unmovable charge +Q along the
trajectory shown. How do the magnitudes of the acceleration at points A, B, and C (which we
denote as aA, aB, aC) compare with each other?
A) aA < aB < aC
B) aC < aB < aA
C) aA = aB = aC
D) aC < aA < aB
E) aC < aA = aB
A
+q
B
C
+Q
The particle’s acceleration is
proportional to the force acting on it.
The Coulomb force gets greater as the charges approach each other.
5. An electroscope (like the one shown on the picture) is negatively charged by
touching it with a negative glass rod. The electroscope leaves spread apart and the
glass rod is removed. Then a positively charged plastic rode is brought closer to the
top of the electroscope, but it does not touch. What happens to the leaves as the
positive rod starts to approach?
A) The leaves get closer together
B) One leaf moves higher, the other lower
C) The leaves spread father apart
D) The leaves don’t move
The positive rod attracts negative charges, so there’s less negative charge left on the
leaves, and they repel each other less.
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6. Two rigid rods with charges attached at the end are arranged as shown. Do the rods repel
each other, attract each other, or neither attract nor repel? (in answering this question, neglect
gravity and friction)
A) repel
B) attract
C) Neither attract nor repel, however if released, the
rods will nevertheless move
D) Neither attract nor repel. If released, the rods
will remain still
E) Impossible to tell from the information given
+Q
−Q
−Q
+Q
The attraction between +Q and -Q beats repulsion between +Q and +Q because they are
closer together.
7. Two large parallel planes are shown edge-on. The left plane has positive charge density η,
while the right plane has negative charge density -η. How do the magnitudes of the electric
field at points A, B, and C compare with each other?
Α) EA > EB >EC
Β) EA < EB < EC
C) EA < EC < EB
D) EA = EB > EC
E) EA < EB = EC
A
B
η
C
−η
The field created by a plane is constant everywhere and is pointed away from (if the plane
is positively charged) or towards (if the plate is negatively charged) the plate. Adding the
fields created by the planes gives the answer E).
8. The picture shows the electric
field lines in some region in
space. Compare the magnitude of
the electric field at points A, B,
and C.
A
B
A) EA > EB > EC
B) EA < EB < EC
C) EA < EC < EB
D) EA = EB > EC
E) EA < EB = EC
The denser the field lines, the stronger is the field.
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C
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9. A uniformly charged rod with linear charge density λ is positioned vertically as shown, so
that the distance between its lower end and the origin is L. What is the magnitude of the
electric field |dE| at the origin (O), created just by a small piece of the rod of length dl
located at a height H above the lower end of the rod?
dl
A) |dE| = k
H 2 + L2
B) |dE| = k
H 2 + L2
y
dl
dλ
C) |dE| = k
H
O
dl
(H 2 + L2 )
x
L
d !E
3
2
Lλ dl
D) |dE| = k
(H 2 + L2 ) 2
E) |dE| = k
λ dl
H 2 + L2
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The charged contained within this piece is λ dl. Its field is given simply by the Coulomb law.
The distance between the piece and the point O is H2+L2. Thus the answer is E).
10. An electric field is given by the expression
!E = 5î − 4 jˆ.
(given in units of N/C). What is the flux of the electric field though the surface shown on the
picture (a horizontal square, sitting in the x-y plane, with a side of one meter long)?
A) 5 N m2/C
B) -4 N m2/C
C) -20 N m2/C
D) 0 N m2/C
E) Impossible to determine from the information given
z
y
The dot product between the field and the normal to this
surface is 0. Thus the answer is zero.
1m
1m
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x
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11. Three positive charges are placed as shown. If released, which way will the charge +q
initially accelerate?
r
2r
A) up
B) down
C) left
D) right
E) the acceleration is zero
up
+q
+2Q
+Q
right
Q/r2 is bigger than (2Q)/(4r)2. Same charges repel each other. Thus the charge will move to
the left.
12. An insulating sphere has been charged throughout its volume, with the charge density ρ.
Subsequently the sphere is broken into two pieces, with the first piece smaller and the second
piece larger. Is the charge density of the first piece ρ1 bigger or smaller than the charge
density of the second piece ρ2?
A) ρ1 > ρ2
B) ρ1 < ρ2
C) ρ1 = ρ2
D) Impossible to tell with the information given
The density does not depend on the size of the pieces, by its very definition.
13. An electron, a particle with a negative charge -e, is injected horizontally into a box bounded
by two charged plates as shown on the picture. The top plate is charged negatively, and the
bottom plate is charged positively. Which
trajectory will the electron follow?
----------------
A
−e
B
C
E
D
+++++++++++
The electron is attracted to the bottom
plate and is repelled from the top one. None of the trajectories are compatible
with electron having no velocity, because then it would go straight down. Since initially the
electron has horizontal velocity (injected horizontally) it must deviate downwards in its
motion, hence D) is the answer.
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14. Four charges are arranged at the vertices of the square as shown. Which way does the electric
field at the center point?
A) Right
B) Up
C) Left
D) Down
E) The field is zero
+Q
r
−2Q
up
r
r
right
−2Q
r
+Q
The fields created by the two +Q charges
cancel each other, and so do the fields created by the two -2Q charges.
15. The electric field throughout a region of space is given by the formula
r
E = Axy ˆi + Bxˆj
where (x,y) are the coordinates of a point in space, and A, B are constants.
r
What is E • ˆj ?
€
A) Axy
€
B) Bx
C) Bx2
D) B
E) none of these
Use the rules of the dot product.
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16. A uniformly-charged insulating rod is bent in the shape of a semi-circle of radius R. The
total charge on the rod is +Q. There are no other charges nearby. What can you say about
the magnitude E of the electric field at the center of the semi-circle?
+Q
E=?
R
A) E= kQ/R2
B) E> kQ/R2
C) E< kQ/R2
D) Not enough information to decide
The magnitude of the fields created by the pieces dQ of the half-ring are all kdQ/R2. Had
they all added together, we would have gotten kQ/R2. But they all point in different
directions and partially cancel each other, so the total must be less than that.
17. A point charge with mass m and positive charge q is suspended on a massless string in a
uniform horizontal electric field of magnitude E. The electric field is adjusted until the string
makes a 45 degree angle with the vertical. Assuming we know E and q, find the mass of the
object. (HINT: draw a free-body diagram.)
E
45◦
+
m, q
A) qE/(g 2 )
B) g/(q E 2 )
C) qE/g
€D) qE 2 / g
€
E) None of these is correct
Draw the free body diagram. Mg should be balanced by T cos 45, while Eq must be
€ balanced by T sin 45. Since cos 45=sin 45, we find that Eq=Mg.
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18. Three charges are placed as shown. If released, the charge at the bottom of the picture will
initially start accelerating in what direction?
A) up
B) down
C) left
D) right
E) the acceleration is zero
r
+Q
r
−Q
up
right
+Q
The charge +Q is attracted to the -Q charge,
but is repelled by the other +Q charge. Both forces
are equal but they point in different directions.
The components of the forces in the vertical directions are equal and opposite, thus they
cancel. The components of the forces in the horizontal direction are equal and are both
directed tot he right. Thus the forces add up to the force pointing to the right.
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Total: 8p
Your Name ______________________________________
Student Number__________________________________
Question 1. A dipole was placed in a box with the electric field lines shown in the figure below.
In which direction will the dipole initially move? (Neglect gravity).
(a) Please carefully CIRCLE the correction direction:
1p
left,
3p
In the space below, please explain
reasoning, briefly but clearly.
right,
up,down,
won’t move at all
!Eyour
The field decreases as we move towards the right, so the positive charges experience
stronger field than the negative charges. Thus the dipole will overall experience a force to
the right (positive charges experience a force which is directed along the field lines, while
negative charges experience an opposite force).
4p
(b) Now the dipole has been rotated slightly, as shown in this figure. In the space below, first
describe the initial (instantaneous) behavior of the dipole after it is released, and then also tell us
what will happen to this dipole as time goes by. Explain your reasoning briefly but clearly.
The dipole will experience a torque which
will rotate it counter clock wise. Simultaneously, it
will move to the right. However, when it rotates
past the vertical position, it will experience a force
pushing it
to the left, so it will slow down and start moving to
the left. When it rotates past the
horizontal position, it will experience a torque in the clock-wise direction, so eventually it
will start rotating clock-wise. It will continue to oscillate left and right, and between the
clock-wise and counter-clock-wise rotations.
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!E
Total: 17p
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Your Name ______________________________________
Student Number__________________________________
Question 2. Two identical conducting balls, A and B,
are attached to insulating stands. The net charge on
ball A is +Q0, and there is no net charge on ball B.
3p
(a). In the diagram at right, sketch the charge
distribution on ball B.
3p
(b). Balls A and B are observed to attract. Use your
sketch (and a brief explanation in the space below)
to account for this attraction.
(a) B will polarize. Its left side will become negative, and its right side will become positive.
(b) The left side of B is negatively charged. These charges are closer to A than the positive
charges on the right side of B. Thus their attraction to A beats the repulsion of the positive
charges.
A third identical conducting ball, ball C, with zero net charge is brought into contact with ball B.
The distance between balls A and B does not change.
1p
(c). Does the attraction between balls A and B
(circle one!)
increase,
3p
decrease,
or remain the same?
Explain.
The positive charge will move from B to C, leaving
B negatively charged. Thus its attraction to A will
increase.
4p
(d).We have redrawn ball C (only) (for your convenience) below: please indicate the direction of
the electric force on ball C by ball A. If this force is zero, state that explicitly. Explain your
reasoning.
C is repelled from A because it is now positively charged.
3p
(e). The three balls are arranged so that they all touch, as shown in the top view diagram. What is
the sign and the magnitude of the charge on ball C? Explain.
The charges will spread over all three balls evenly because the charges all
repel each other, so it will be +Q/3.
B
A
C
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