8.02
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
Spring 2013
Problem 1 (25 points): answers without work shown will not be given any credit.
A uniformly charged slab of charge per unit volume !
,
extends from x
= !
2 m to x
= + 2 m (see upper figure). A uniformly charged sheet of charge per unit area !
is located at x
= + 2 m . The electric potential
V
( x ) for this distribution of charge is plotted in the lower figure
Analytically, the potential function is given by expression
V ( x ) =
#
%
$
%
&
!
(1 V " m
(1 V " m
(1 V " m
!
2
!
1
!
1
)
) x x
) x !
6 V ; x < !
2 m
2
+
+ (3 V
6 V ; x
" m
>
!
1 ) x
2 m
!
2 V ; !
2 m < x < + 2 m . a) Determine a vector expression for the electric field in the regions: (i)
x
(ii) !
2 m < x < + 2 m , and (iii)
x
> 2 m .
< !
2 m b) Determine !
, the charge per unit volume in the slab. Your answer can include
,
!
0
. [Hint: use Gauss’s Law]. c) Determine !
, the charge per unit area on the sheet. Your answer can include !
0
[Hint: use Gauss’s Law].
.
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2

Problem 2 (25 points): answers without work shown will not be given any credit.
A negatively charged particle of mass m and charge !
q moving upward with velocity v = V
ˆ j
is at the origin at t
= 0 and
. Its subsequent trajectory is shown in the sketch.
The particle moves first for y > 0 in a semi-circle in the
xy moves for y < 0 in a semi-circle in the
yz when y = 0 same, although the direction of v
-plane with radius
-plane with radius
R
again (see sketch). The magnitude of the velocity
/ 2
V =
R . It then
, arriving at point v is always the
P
changes in time. For the region y > 0 , the magnetic field is uniform with magnitude magnitude is
B
2
B
1
. For y < 0 , the magnetic field is also uniform but the
. Ignore any gravitational field.
(a) What is the direction of the magnetic field for the region y > 0 ?
(b) Derive an expression for the magnitude
B
1
of the magnetic field for the region in terms of the given quantities, that is in term of
q
, m , R , and
(c) What is direction of the magnetic field in the region y < 0 ?
V . y
>
0
(d) What is the magnitude
B 2
V ?
of the magnetic field for y < 0 in terms of
q
, m , R
(e) How long does it take the charged particle to move from the origin to point P located as shown in the figure above? Give your answer in terms of the given
, m , R , and V as needed.
, and quantities
q
3

(f) When the particle passes point P into the region y
>
0 , it now sees not only the magnetic field above in (a) and (b) for y > 0 , but we also turn on a uniform electric field, such that the particle now moves vertically upwards in a straight line, as shown in the sketch. What is the magnitude and direction of this electric field? Express your answer in terms of in terms of the given quantities, that is in term of
q and V .
, m , R ,
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5

Problem 3 (25 points): answers without work shown will not be given any credit.
A certain coaxial cable consists of a copper wire, radius a , surrounded by a concentric copper tube of inner radius c . The space between is partially filled (from a out to b ) with a material of dielectric constant !
. The space between b and c is empty (vacuum).
The cable has length L
/ L ) of this cable. a) Determine the capacitance per unit length ( C b) When the conducting surfaces of the capacitor are connected to a battery with a potential difference V , determine an expression for the stored energy per unit length ( U / L ) . c) While the battery is still connected, the entire region between the cylindrical shells is filled with the material of dielectric constant !
. By what amount has the magnitude of the free charge per unit length changed on either conducting shell
!
( Q / L ) " ( Q f
# Q i
) / L ?
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Problem 4 of 4: (25 points) Concept Questions (Parts A through E)
Part A (5 points). A parallel plate capacitor, initially with vacuum between the plates, is connected to a battery and charged to + Q on the upper plate and !
Q on the lower plate respectively. The battery is then disconnected and afterwards the area of each plate is decreased by the same amount.
Which of the following statements is true? a) The electric field in the capacitor increases and the energy stored increases. b) The electric field in the capacitor remains the same and the energy stored increases. c) The electric field in the capacitor decreases and the energy stored increases. d) The electric field in the capacitor increases and the energy stored remains the same. e) The electric field in the capacitor remains the same and the energy stored remains the same. f) The electric field in the capacitor decreases and the energy stored remains the same. g) The electric field in the capacitor increases and the energy stored decreases. h) The electric field in the capacitor remains the same and the energy stored decreases. i) The electric field in the capacitor decreases and the energy stored decreases.
The correct answer is a.
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10

Part B (5 points) a) the surface charge at O2 is positive and
V
O 1
.
Two hollow conducting shells are concentric. The outer shell 2 is negatively charged with charge
V
O 1
!
Q . The inner shell is positively charged with charge a negatively charged point-like object with charge be the potential at O1 and
V
O 2
!
Q
+ 2 Q . We then place
at the center of the shells. Let be the potential at O2. After equilibrium is reached,
> V
O 2 b) the surface charge at O2 is positive and
V O 1
< V
O 2
. c) the surface charge at O2 is positive and
V O 1
= V
O 2
. d) the surface charge at O2 is negative and
V O 1
> V
O 2
. e) the surface charge at O2 is negative and
V
O 1
< V
O 2
. f) the surface charge at O2 is negative and
V
O 1
= V
O 2
. g) the surface charge at O2 is zero and
V
O 1
> V
O 2
. h) the surface charge at O2 is zero and
V O 1
< V
O 2
.
= V
O 2
. i) the surface charge at O2 is zero and
V
O 1
The correct answer is g.
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12

Part C (5 points)
A long wire is formed into a semi-circle with center at point P and two long straight segments joined to the semi-circle, as shown. The long segments are perpendicular to the plane of the semi-circle and are parallel. The wire carries current I .
If we compare the magnitude of the total field at point P in the figure to the magnitude of the field due to the semi-circle segment alone, a) The magnitude of the total field is less than that of the semi-circular segment alone. b) The magnitude of the total field is equal to that of the semi-circular segment alone. c) The magnitude of the total field is greater than that of the semi-circular segment alone. d) Impossible to determine without more information.
The correct answer is b.
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Part D (5 points)
Two charged particles of identical charge move in circular orbits in the same uniform magnetic field
B
= B o
. The radius of the orbit of particle 1 is twice the radius of the orbit of particle 2. Particle 2 has twice times the mass of the particle 1, m
2
=
2 m
1
. a) The speed of particle 2 is four times the speed of particle 1. b) The speed of particle 2 is two times the speed of particle 1. c) The speed of particle 2 is equal to the speed of particle 1. d) The speed of particle 2 is ½ of the speed of particle 1. e) The speed of particle 2 is ¼ of the speed of particle 1
The correct answer is e.
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Part E (5 points)
At the instant shown in the figure to the right, an electron at the position shown with charge velocity v =
q v
ˆ j
= !
e is moving with
. A very long wire carries a current in the direction shown in the figure. At the instant depicted, the force on the electron is
1.
Zero.
2.
in the + ˆ i -direction.
3.
in the !
ˆ i -direction.
4.
in the + j
ˆ
-direction.
5.
in the !
j
ˆ
-direction.
6.
in the + ˆ -direction.
7.
in the !
ˆ
-direction.
The correct answer is 2.
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