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# SAI Examen 1 2012Spr

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```MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
8.02
Spring 2012
Exam One Equation Sheet
!
qq
F12 = ke 1 2 2 r̂12 , force by 1 on 2
r
! 12
F = qE
! !
q
q!
E(r) = ke 2 r̂ = ke 3 r
r
r
N
! !
q
! &quot;
E(r) = ke &quot; ! i&quot; 3 (r ! ri )
i=1 r ! r
i
! !
! !
dq
dq&quot;(r # r &quot; )
E(r) = ke ! 2 r̂ = ke !
! ! 3
r # r&quot;
source r
source
! ! q
E ! d A = enc
#o
closed surface
&quot;
&quot;&quot;
!
dA points from inside to outside
B !
!
!V AB = VB &quot; V A = &quot; \$ E # d s
A
V (r) ! V (&quot;) = ke
q
r
N
q
!
V (r) ! V (&quot;) = ke # ! i &quot;
i=1 r ! ri
!
V (r) ! V (&quot;) = ke
\$
source
dq#
! !
r ! r#
!U = q!V
q!V + !K = 0
U stored = ke !
all pairs
qi q j
rij
; U (&quot;) = 0
Electric Dipole
! N &quot;
p = ! qi ri
i=1
!
p # r̂
!
V (r) ! V (&quot;) = ke 2
r
!
!
E = !&quot;V
&quot;V
&quot;V
&quot;V
Ey = !
Ex = !
Ez = !
&quot;y
&quot;x
&quot;z
&quot;V
Er = !
for spherical and cylindrical
&quot;r
symmetry
1 #V
E! = &quot;
for cylindrical symmetry
r #!
Constants
ke =
1
= 9 # 109 N \$ m 2 \$ C%2
4!&quot; 0
Circumferences, Areas, Volumes
1) The area of a circle of radius r is
! r 2 . Its circumference is 2! r .
2) The surface area of a sphere of
radius r is 4! r 2 . Its volume is
(4 / 3)! r 3 .
3) The area of the sides of a cylinder
of radius r and height h is 2! rh .
Its volume is ! r 2 h .
1
2
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
8.02
Spring 2012
Exam One
Your Table and Group (e.g. 10A): ______________
____L01 Figueroa
MW 10-12 am
____L05 Gore
TR 11-1 pm
____L02 Tegmark
MW 12 -2 pm
____L06 Rajagopal TR 1-3 pm
____L03 Soljacic
MW 2 -4 pm
____L07 Belcher
TR 3-5 pm
____L04 Dourmashkin TR 9 -11 am
____L08 Stevens
MW 10-12 pm
____L09 Chen
MW 2-4 pm
Score
Problem 1 (25 points)
Problem 2 (25 points)
Problem 3 (25 points)
Problem 4 (25 points)
TOTAL
3
Problem 1 (25 points)
In this problem you are asked to answer 5 questions, each worth 5 points.
Question 1 (5 points) Two small objects each with a charge of +Q exert a force of
!
magnitude F ! F on each other.
We replace one of the objects with another whose charge is +4Q , and we move the
+Q and +4Q charges to be 3 times as far apart as they were:
!
What is the magnitude of the force F1 ! F1 on the +4Q charge above?
a) F1 = F / 9
b) F1 = F / 3
c) F1 = 4F / 9
d) F1 = 4F / 3
e) None of the above.
4
Question 2 (5 points)
Three infinite sheets of charge lying in the yz -plane are shown in the sketch. The sheet
on the right at x = d is positively charged with charge per unit area of +! , the sheet in
the middle at x = 0 is positively charged with charge per unit area of +4! , and the sheet
on the left at x = !d is negatively charged with charge per unit area of !&quot; .
What is the electric field at the point P in the region x &gt; d ?
a) zero.
! !
b) E = î .
&quot;0
!
&quot;
c) E = ! î .
#0
! 2!
d) E =
î .
&quot;0
!
2&quot;
e) E = !
î .
#0
! 4!
f) E =
î .
&quot;0
!
4&quot;
g) E = !
î .
#0
h) None of the above.
5
Question 3 (5 points)
We have two electric dipoles. Each dipole consists of two equal and opposite point
charges at the ends of an insulating rod of length d . The dipoles sit along the x -axis a
distance r apart, oriented at right angles to each other as shown below. Their separation
r &gt;&gt; d . You may take d to be small but not zero.
The dipole ON RIGHT
a) will feel a force to the left and a torque trying to make it rotate clockwise.
b) will feel a force to the left and a torque trying to make it rotate counterclockwise.
c) will feel a force to the left and no torque.
d) will feel a force to the right and a torque trying to make it rotate clockwise.
e) will feel a force to the right and a torque trying to make it rotate counterclockwise.
f) will feel a force to the right and no torque.
g) will feel no force and a torque trying to make it rotate clockwise.
h) will feel no force and a torque trying to make it rotate counterclockwise.
i) will feel no force and no torque.
6
Question 4 (5 points)
A thin rod extends along the x -axis from x = !l / 2 to x = l / 2 . The rod carries a
uniformly distributed positive charge +Q . Consider a point P lying in the z = 0 plane
with coordinates (x, y,0) . Which of the following expressions describes the electric
potential difference V (P) ! V (&quot;) , between infinity and the point P ?
a)
keQ x # = l / 2 ((x ! x # ) î + yĵ) dx #
.
V (P) ! V (&quot;) =
l x # =\$! l / 2 ((x ! x # )2 + y 2 )3/ 2
b)
V (P) ! V (&quot;) =
keQ x # = l / 2 ((x ! x # ) î + yĵ) dx #
.
l x # =\$! l / 2 ((x ! x # )2 + y 2 )1/ 2
c)
V (P) ! V (&quot;) =
keQ x # = l / 2
(x ! x # ) dx #
.
\$
l x # = ! l / 2 ((x ! x # )2 + y 2 )3/ 2
d)
keQ x # = l / 2
(x ! x # ) dx #
.
V (P) ! V (&quot;) =
\$
l x # = ! l / 2 ((x ! x # )2 + y 2 )1/ 2
e)
keQ x # = l / 2
dx #
.
V (P) ! V (&quot;) =
\$
l x # = ! l / 2 ((x ! x # )2 + y 2 )3/ 2
f)
keQ x # = l / 2
dx #
.
V (P) ! V (&quot;) =
\$
l x # = ! l / 2 ((x ! x # )2 + y 2 )1/ 2
g) None of the above.
7
Question 5 (5 points)
!
The area vector dA at each point on a
closed surface (i.e., a surface that
surrounds a volume) are always chosen
to point out of the enclosed volume. A
closed imaginary surface is called a
Gaussian surface. The Gaussian surface
below is a cylinder. A positive charge is
located on the cylinder axis above the
Gaussian cylinder, as shown on the
figure to the right.
Which statement is correct about the flux
&quot;&quot;
! !
E ! d A through surface B and
surface B
through the entire closed surface A + B + C ?
a) The flux through surface B is positive and through the entire closed surface it
is positive.
b) The flux through B is positive and through the entire surface it is negative.
c) The flux through B is positive and through the entire surface it is zero.
d) The flux through B is negative and through the entire surface it is zero.
e) The flux through B is negative and through the entire surface it is negative.
f) The flux through B is negative and through the entire surface it is positive.
8
Problem 2 (25 points)
NOTE: YOU MUST SHOW WORK in order to get any credit for this problem.
Three charges equal to +Q , !2Q , and
+Q are placed on the vertices of an
equilateral triangle with sides a, as shown
in the sketch. The point P is at the
origin. Note that sin(60! ) = 3 / 2 and
cos(60! ) = 1 / 2 .
!
a) What is the x-component of the electric field E at point P ?
!
b) What is the y-component of the electric field E at point P ?
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c) What is the electric potential V at point P, assuming that V (!) = 0 ?
d) Suppose you now move a point charge with charge +5Q from infinity to point P
in the above problem. How much work does it take you to do this?
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e) What is the dipole moment vector of the original set of three charges?
f) Suppose the original three charges were glued to a wooden equilateral triangle
which was free to rotate
! about its center, and then that triangle was placed in a
uniform electric field E = Eo ŷ , where Eo &gt; 0 . Will the triangle rotate
counterclockwise or clockwise?
11
Problem 3 (25 points)
NOTE: YOU MUST SHOW WORK in order to get any credit for this problem.
A very thin disk of radius R is positively charged with uniform charge per unit area +! .
Consider a point P lying a distance x &gt; 0 from the center of the disk along the axis of
symmetry of the disk.
Integration formulas you may find useful:
x2 &quot;
dr !
r!
= 2
+C
2 3/ 2
(x + r ! )
(x + r ! 2 )1/ 2
&quot; (x
2
&quot; (x
r !dr !
1
=
#
+C
2
+ r ! 2 )3/ 2
(x 2 + r ! 2 )1/ 2
&quot; (x
dr !
= ln r ! + (x 2 + r ! 2 )1/ 2 + C
2
2 1/ 2
+ r! )
2
r !dr !
= (x 2 + r ! 2 )1/ 2 + C
+ r ! 2 )1/ 2
a) What is the electric potential difference V (P) ! V (&quot;) ? Hint: you may find the
integration area element da! = 2&quot; r !dr ! useful.
12
b) What is the direction and magnitude of the electric field at the point P ?
13
c) Find an expression for the direction and magnitude of the electric field when
x &lt;&lt; R .
d) How does your result in part c) compare to the electric field of a uniformly
charged infinite plane with charge per area +! ? Explain your reasoning.
14
15
Problem 4 (25 points)
NOTE: YOU MUST SHOW WORK in order to get any credit for this problem.
A very long solid cylinder of radius R1 carries a non-uniform volume charge density
! = Cr ; 0 &lt; r &quot; R1 , where C is a positive constant with units [C ! m &quot;4 ] . It is surrounded
by a very thin negatively charged cylindrical shell of radius R2 with uniform charge per
area ! &lt; 0 as shown in the figure. The electric field is zero for r &gt; R2 . You may assume
the cylinders are infinite.
a) Find an expression for the charge per area ! on the outer shell in terms of C , R1 ,
and R2 as needed.
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b) Find a vector expression for the electric field in the regions (i) 0 &lt; r ! R1 and (ii)
R1 &lt; r &lt; R2 . For each region clearly show your choice of Gaussian surface.
Express your answers in terms of r , C , R1 , and R2 as needed.
17
c) A positively charged object of charge Q &gt; 0 and mass M is released from rest at
r = R1 . What is the speed of the object when it reaches r = R2 ? Express your
answers in terms of r , C , Q , M , R1 , and R2 as needed.
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