Physics 212
Exam #1
NAME _______________________
Write down your name also on the back of the
final page of this exam.
-12 2
C /N.m2
Use o= 8.85 x 10
9
, o= 9 x 10 N.m2/C2
, g = 10 m/s2
1. The figure below shows two particles fixed in place: a particle of positive charge
+18e at the origin, and a particle of negative charge -2e at 6 cm far away. Here e is
the magnitude of the charge of an elementary charge.
1A At what point along the axis that joins the charges (other than infinite far away)
can a proton
3 cm
be placed so
+ 18e
- 2e
that it is in
x
equilibrium?
Indicate your
6 cm
answer with
an “x” drawn in the graph provided above. Indicate explicitly also the distance
between the proton and the closest charge
1B Draw the electric fields that exist i) at 1 cm to the left and ii) at 1 cm to the right
of the proton’s
3 cm
position found
+ 18e
- 2e
in part 1A
x
above.
6 cm
______________________________________________________________________________
2. The figure shows a particle of positive charge q
= +10-3 Coulomb, placed at the origin.
2A Rank the positions P, S and U according to
the magnitude of the electric filed existent at
those points. Greatest first.
a) P, S, U,
b) U, S=P c) P, U=S
d) U, P, S
e) NA
Z
U
(2.5, 0, 2.5)
X
P (0,4,3)
Y
+q
S (5,5,0)
Coordinates given in units of cm.
2B Find the electric field at point P (in units of
109 N/C)
a) (0, 2.9, 2.2)
b) (-3.1, -3.1, 0)
d) (3, 0, 4)
e) NA
c) (0, 4.4, 5.8)
______________________________________________________________________________
3. The figure below shows a compact sphere of radius R=6 cm with charge Q = + 9 C
uniformly distributed through its volume
3.A In the diagram provided below, sketch the magnitude of the electric field E(r)
established by the uniformly
distributed charge distribution
E(r)
inside and outside the sphere
R
r
R
3.B The electric field E at the point (3 cm, 4 cm, 0) is equal to (in units of 107 N/C)
a) (1.9, 1.9, 0)
b) 0.6, 0.8, 0)
c) (1.1, 1.5, 0)
d) (0.4, 0.3, 1)
e) NA
_______________________________________________________________________
4. 4A The figures show two uniformly
charged spheres. The positive
charge on sphere B is 2 times
the positive charge on sphere A.
Which force diagram correctly
shows the magnitude and
direction of the electrostatic
forces acting on each sphere?
a)
b)
c)
d)
e)
f)
a)
+++
++
++
+++
B
b)
+++
++
++
+++ B
d)
+
++
++
+++ B
f)
A
c)
A
e)
+
A +
+++
++
++
+++
+++
++
++
+++ B
++
++
++
+++B
A
A
A +
B
4B The figure shows two point charges of charge +3q and – q respectively are located along
long the x-axis and equidistance from the origin. Sketch, the electric field lines
everywhere around the
charges AND the equipotential
surfaces
passing through A, B
and C, respectively.
(Point B is the middle
A
C
point
between
the
B
charges).
-q
+ 4q
5. The figure below shows a parallel-plate capacitor. In what follows, assume for
simplicity that the electric field around the center of the plates is equivalent to the
electric field produced by infinite plates charged uniformly all over their surface.
In the figures, Q stands for the initial charge accumulated in the top plate, E is the
electric field in the central region between the plates, and d is the distance between
the plates.
5A While keeping the battery connected to the plates, an external agent (not shown in
the figure) increases the distance between the plates
+
Ei
2V -
Qi
+
di
2V -
At t=0
Qf
Ef
df
At t > 0
Which of the following expression(s) describe(s) correctly what happens as the
distance between the plates is increased?
a) The electric field increases.
b) The magnitude of the charge in each plate increases.
c) The electric field remains constant.
d) The magnitude of the charge in each plate decreases.
e) The electric potential between the plates decreases.
5B In this case the capacitor is initially charged and then the battery is disconnected.
Ei
At t=0
Qi
Qf
di
Ef
df
At t > 0
Which of the following expression(s) describe(s) correctly what happens as the
distance between the plates is increased?
a) The electric field increases.
b) The electrical potential energy stored in the capacitor decreases.
c) The electric field remains constant.
d) The magnitude of the charge in each plate decreases.
e) The electric potential between the plates decreases.
6. 6A A graph of the electric potential as a function of x is given in the figure below.
Plot (on scale) the electric field component Ex as a function of x (use the rightside diagram). Ignore the points on the graph where the slop changes
discontinuously.
8
E(x)
8
V(x)
4
6
v (x ) V o lt s
v (x ) V o lt s
6
x
4

2
0

2
1 cm
1 cm
x
0
A
B
C
D
E
G
F
H
x
0
I
A
B
C
D
E
F
G
H
I
6B Two charges, one positive (q) and the other negative (-q) are located along the zaxis, and separated by a constant
distance “d”. This charge configuration
Z
r P (x,y,z)
is called a dipole. For a location, such as
point P in the figure, very far away from
+q
the dipole, the electric potential is given
d
Y
by
-q
V ( x , y, z ) 
X
2
2
2 1/ 2
r  (x  y  z )

1
qd z
2
2
2 3/ 2
4  0 ( x  y  z )
1
qd z
4  0
r3
Which of the following expressions gives the z-component of the electric field?
a)
qd 3z 2  r 2
4 0
b)
r5
d)
qd z  r
4 0 r 3 / 2
qd 3 xz
c)
4 0 r 5
e)
qd 3 zy  r 2
4 0
qd z 2  r 2
4 0
r5 / 2
r3 / 2
7. 7A In the figure shown below, the magnitude of the electric field at point “P” due to
the four point charges is equal to (in units of N/C):
a) 4.4 x 108
b) 3.58 x 108
c) 0.4 x 108
d) 0
e) NA
7B In the figure shown below, what is the electric potential at point “P”, if V=0 at
infinity (in units of Volt)?
a) 1.32 x 107
b) 2.4 x 106
d
+7q
d d
P d
c) - 1.2 x 106
d) 0
e) NA
-24q
+6q
q=1C
d= 3 cm
+7q
8. 8A How much energy (in Joules) is stored in the system shown in the figure?
q3
h
h
q1
h = 10 cm
r = 6 cm
r
q2
q1= +10 C
q2= -20 C
q3= +30 C
a) 23
b) 17
c) 57
d) 71
e) NA
8B Starting with the configuration given above, how much net work must an external
agent do to exchange the positions of
i) q1 and q2 , and ii) q1 and q3, respectively?
a) 0, 0
b) ≠ 0, ≠ 0
c) 0, ≠ 0
e) There is not enough information to decide.
d) ≠ 0, 0
9. 9A Find the equivalent capacitance of the
circuit shown, where C1= 1 F, C2 =
2 F and C3 = 1/3 F:
a) 3 / 10 F
b) 2 F
c) 5 F
d) 8 / 3 F
e) NA
9B Find the equivalent capacitance of the
circuit shown, where C1= 1 F, C2 =
2 F and C3 = 1/3 F:
a) 10/3 F
b) 2 F
c) 4/5 F
d) 8/3 F
C3
+
C2
C1
_
V
C2
_+
C3
V
C1
e) NA
______________________________________________________________________________
10
10A The figure below shows plots of charge versus potential difference for three
parallel-plate capacitors, which have the plate areas and separations given in
the table. Which of the plots goes with which of the capacitors? Give your
answer by filling out the “Graph” column in the table
q
Capacitor Area Separation
Graph
1
A
d
2
2A
d
b
a
3
A
2d
c
a
b
c
V
10B The voltage across the capacitor C2, before and after introducing a dielectric
slab (k = 5) are, respectively:
a) 3V, 3 V
b) 2V, 1V
C1=1 F
c) 3V, 1V
d) 1V, 1V
_+
V = 6 Volts
e) NA
C2= 1 F
k=5
BONUS QUESTION (3 points)
B1. The figure shows two disks and a flat ring, each with the same uniform charge Q. Rank the
objects according to the magnitude of the electric field they create at point A (which are at
the same vertical height); greatest first.
A
A
R
A
2R
P
S
R
2R
U
Answer___P, S, U________________
B2. What are the units of Volts in MKS?
Answer____Kg m2/ (C s2)___________________
Some formulas:




nano

1 q 2 q1 
u
Coulomb's Law: F 
4 0 r 2
Electric field, along the z-axis, due to a charge Q distributed uniformly along a thin ring of
radius R.:


E
1

Qz
4 0 z  R
2

2 3/ 2
kˆ
For an infinite uniformly charged sheet:
E =  /2o

Gauss' Law

 = E  ds = q /o

, where q is the net charge inside the gaussian surface
S

Definition of Electric Potential V (r ) 

Potential difference
W ext ( 
r)
q
0
q0


V  V   E  d s
B
B
A
A

Electric potential due to a point charge q:
V 
1 q
4 r
0

Relationship between E and V:

About capacitance
Ex = - dV/dx
Q=CV
For a parallel-plate capacitor C = Ao /d
U = CV2 / 2 = Q2 / 2 C
Capacitors connected in parallel Cequiv = C1 + C2 + C3
Capacitors connected in series 1/Cequiv = (1/C1) + (1/C2) +(1/C3)
When filing a capacitor with a dielectric: Capacitance increases by a factor of k

 sin  d   cos
 Cos d  sin 