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Physics 226
Fall 2013
Problem Set #1
NOTE: Show ALL work and ALL answers on a piece of separate loose leaf paper, not on this sheet
.
Due on Thursday, August 29th
1)
Skid
and Mitch are pushing on a sofa in opposite
directions with forces of 530 N and 370 N respectively.
The mass of the sofa is 48 kg. The sofa is initially at rest
before it accelerates. There is no friction acting on the
sofa. (a) Calculate the acceleration of the sofa. (b) What
velocity does the sofa have after it moves 2.5 m? (c) How
long does it take to travel 2.5 m?
2) You have three force
vectors acting on
a
mass at the origin.
Use the component
method we covered
in lecture to find
the magnitude an
d
direction of the re-
sultant force acting
on the mass.
3) You have three force
vectors acting on a
mass at the origin.
Use the component
method we covered
in lecture to find
the magnitude and
direction of the re-
sultant force acting on
the mass.
4) A bowling ball rolls off of a table that is 1.5 m tall. The
ball lands 2.5 m from the base of the table. At what speed
did the ball leave the table?
5) Skid throws his guitar up
into the air with a velocity
of 45 m/s. Calculate the
maximum height that the
guitar reaches from the point
at which Skid lets go of the
guitar. Use energy methods.
6)
A beam of mass 12 kg and length 2 m is attached to a
hinge on the left. A box of 80 N is hung from the beam
50 cm from the left end. You hold the beam horizontall
y
with your obviously powerful index finger. With what
force do you push up on the beam?
Mitch Sofa Skid
7) The tennis ball of mass 57 g which
you have hung in your garage that
lets you know where to stop your
car so you don’t crush your garbage
cans is entertaining you by swinging
in a vertical circle of radius 75 cm.
At the bottom of its swing it has a
speed of 4 m/s. What is the tension
in the string at this point?
y
8) Derivatives:
a) Given: y = (4x
+
L)(2×2 – L), find
dx
dy
.
b) Given:
Lx2
Lx2lny , find
dx
dy
.
9) Integrals:
a) Given:
o
o
45
45
d
r
cosk
, evaluate.
b) Given:
R
0 2322
dr
xr
kxr2 , evaluate.
ANSWERS:
1) a) 3.33 m/s2
b) 4.08 m/s
c) 1.23 s
2) 48.0 N, 61.0º N of W
3) 27.4 N, 16.1º S of E
4) 4.52 m/s
5) 103.3 m
6) 78.8 N
7) 1.78N
8) a) 24×2 + 4xL – 4L
b) 22 x4L
L4
9) a)
r
k2
b)
22 xR
x1k2
F2 = 90 N
F1 = 40 N 35
45 x
F3 = 60 N
y
F1 = 45 N 60
F2 = 65 N
50 x
70
F3 = 85 N
Guitar
Skid
Physics 226
Fall 2013
Problem Set #2
1) A plastic rod has a charge of –2.0 C. How many
electrons must be removed so that the charge on the rod
becomes +3.0C?
–
+
+
+
2)
Three identical metal spheres, A, B, and C initially have
net charges as shown. The “q” is just any arbitrary amount
of charge. Spheres A and B are now touched together and
then separated. Sphere C is then touched to sphere A and
separated from it. Lastly, sphere C is touched to sphere
B
and then separated from it. (a) How much charge ends
up on sphere C? What is the total charge on the
three spheres (b) before they are allowed to touch each
other and (c) after they have touched? (d) Explain the
relevance of the answers to (b) and (c).
3)
Skid of 40 kg and Mitch of 60 kg are standing on ice on
opposite sides of an infinite black pit. They are each
carrying neutral massless spheres while standing 8 m
apart. Suppose that 3.0 x 1015 electrons are removed from
one sphere and placed on the other. (a) Calculate the
magnitude of the electrostatic force on each sphere. Are
the forces the same or different? Explain. (b) Calculate
the magnitude of the accelerations for Skid and Mitch at
the moment they are 8 m apart. Are they the same or
different? Explain. (c) As Skid and Mitch move closer
together do their accelerations increase, decrease, or
remain the same? Explain.
4) An electron travels in a circular orbit around a stationary
proton (i.e. a hydrogen atom). In order to move in a circle
there needs to be a centripetal force acting on the electron.
This centripetal force is due to the electrostatic force
between the electron and the proton. The electron has a
kinetic energy of 2.18 x 10–18 J. (a) What is the speed of
the electron? (b) What is the radius of orbit of the
electron?
5)
Three charges are arranged as shown. From the left to
the right the values of the charges are 6 μC, – 1.5 μC, and
– 2 μC. Calculate the magnitude and direction of the net
electrostatic force on the charge on the far left.
6) For the same charge distribution of Problem #5, calculate
the magnitude and direction of the net electrostatic force
on the charge on the far right.
7)
Two charged spheres are connected to a spring as shown.
The unstretched length of the spring is 14 cm. (a) With
Qa = 6 μC and Qb = – 7 μC, the spring compresses to an
equilibrium length of 10 cm. Calculate the spring
constant. (b) Qb is now replaced with a different charge
Qc. The spring now has an equilibrium length of 20 cm.
What is the magnitude of the charge Qc? (c) What is the
sign of Qc? How do you know this?
8)
The two charges above are fixed and cannot move. Find
the location in between the charges that you could put a
proton so that the proton would have a net force of zero.
9) Three charges are fixed to an xy coordinate system.
A charge of –12 C is on the y axis at y = +3.0 m.
A charge of +18 C is at the origin. Lastly, a charge of
+ 45 C is on the x axis at x = +3.0 m. Calculate the
magnitude and direction of the net electrostatic force on
the charge x = +3.0 m.
10) Four charges are situated
at the corners of a square
each side of length 18 cm.
The charges have the same
magnitude of q = 4 μC but
different signs. See diagram.
Find the magnitude and
direction of the net force on
lower right charge.
+5q – 1q Neutral
C B
A
Skid Mitch
Infinite
Black Pit
– –
3 cm 2 cm
+
– +
Qa Qb
+
8 cm
+
4 μC 12 μC
11) For the same charge distribution of problem #10, find
the magnitude and direction of the net force on upper
right charge.
20
12)
All the charges above are multiples of “q” where q = 1μC.
The horizontal and vertical distances between the charges
are 15 cm. Find the magnitude and direction of the net
electric force on the center charge.
13) Use the same charge distribution as in problem #12 but
change all even-multiple charges to the opposite sign.
Find the magnitude and direction of the net electric force
on center charge.
14) Two small metallic spheres, each
of mass 0.30 g, are suspended by
light strings from a common point
as shown. The spheres are given
the same electric charge and it is
found that the two come to
equilibrium when the two strings
have an angle of 20 between
them. If each string is 20.0 cm
long, what is the magnitude of the
charge on each sphere?
– 4q +9q +4
q
+3q +3q +8q
15)
+6q R2 – 4q 12 cm m
A meter stick of 15 kg is suspended by a string at the
60 cm location. A mass, m, is hung at the 80 cm mark.
A massless charged sphere of + 4 μC is attached to the
meter stick at the left end. Below this charge is another
charge that is fixed 12 cm from the other when the meter
stick is horizontal. It has a charge of – 4 μC. Calculate
the mass, m, so that the meter stick remains horizontal.
ANSWERS:
7) a) 945 N/m
b) 4.2 x 10–5
C
8) 2.93 cm
9) 0.648 N, 17.2º
10) 4.06 N,
45º
11) 6.66 N, 64.5º
12) 19.69 N, 80.1º
13) 18.5 N, 23.4º
14) 1.67 x 10–8 C
15) 10.56 kg
1) 3.1 x 1013 e
–
2) a) +1.5q
b) +4q
c) +4q
3) a) FE, Skid = 32.4 N
b) aSkid = 0.81 m/s2
4) a) 2.19 x 106 m/s
b) 5.27 x 10–11 m
5) FE = 133.2 N, →
6) FE = 24.3 N, →
Physics 226
Fall 2013
Problem Set #3
1) A charge of –1.5 C is placed on the x axis at
x = +0.55 m, while a charge of +3.5 C is placed at
the origin. (a) Calculate the magnitude and direction of
the net electric field on the x-axis at x = +0.8 m.
(b) Determine the magnitude and direction of the force
that would act on a charge of –7.0 C if it was placed on
the x axis at x = +0.8 m.
2) For the same charge distribution of problem #1, do the
following. (a) Calculate the magnitude and direction of
the net electric field on the x-axis at x = +0.4 m.
(b) Determine the magnitude and direction of the force
that would act on a charge of –7.0 C if it was placed on
the x axis at x = +0.4 m.
3)
Charges are placed at the three corners of a rectangle as
shown. The charge values are q1 = 6 nC, q2 = – 4 nC, and
q3 = 2.5 nC. Calculate the magnitude and direction of the
electric field at the fourth corner.
4) For the same charge distribution of problem #3, with the
exception that you change both q1 and q2 to the opposite
sign, calculate the magnitude and direction of the electric
field at the fourth corner.
5) A drop of oil has a mass of 7.5 x 10–8 kg and a charge of
– 4.8 nC. The drop is floating close the to Earth’s surface
because it is in an electric field. (a) Calculate the
magnitude and direction of the electric field. (b) If the
sign of the charge is changed to positive, then what is the
acceleration of the oil drop? (c) If the oil drop starts from
rest, then calculate the speed of the oil drop after it has
traveled 25 cm.
6) A proton accelerates from rest in a uniform electric field
of magnitude 700 N/C. At a later time, its speed is
1.8 x 106 m/s. (a) Calculate the acceleration of the proton.
(b) How much time is needed for the proton to reach this
speed? (c) How far has the proton traveled during this
time? (d) What is the proton’s kinetic energy at this
time?
7)
All the charges above are multiples of “q” where
q = 1μC. The horizontal and vertical distances between
the charges are 25 cm. Find the magnitude and direction
of the net electric field at point P.
8) Use the same charge distribution as in problem #7 but
change all even-multiple charges to the opposite sign.
Find the magnitude and direction of the net electric field
at point P.
9)
In the above two diagrams,
M
& S, an electron is given an
initial velocity, vo, of 7.3 x 106 m/s in an electric field of
50 N/C. Ignore gravitation effects. (a) In diagram M,
how far does the electron travel before it stops? (b) In
diagram S, how far does the electron move vertically after
it has traveled 6 cm horizontally? (Hint: Think projectile
motion)
– +
+
P
q3 q2
q1
35 cm
20 cm
– 8q
– 4q
+9q
+9q
– 5q
+6q +6q
+2q
P
– – vo vo
M S
10) A 2 g plastic sphere is suspended
by a 25 cm long piece of string.
Do not ignore gravity. The sphere
is hanging in a uniform electric
field of magnitude 1100 N/C. See
diagram. If the sphere is in
equilibrium when the string makes
a 20 angle with the vertical, what
is the magnitude and sign of the
net charge on the sphere?
11) You have an electric dipole of
opposite charges q and distance 2a
apart. (a) Find an equation in terms
of q, a, and y for the magnitude of
the total electric field for an electric
dipole at any distance y away from
it. (b) Find an equation in terms of
q, a, and y for the magnitude of the
total electric field for an electric
dipole at a distance y away from it
for when y >> a.
12)
A dipole has an electric dipole moment of magnitude
4 μC·m. Another charge, 2q, is located a distance, d,
away from the center of the dipole. In the diagram all
variables of q = 20 μC an
d d
= 80 cm. Calculate the net
force on the 2q charge.
13) An electric dipole of charge 30 μC and separation
60 mm is put in a uniform electric field of strength
4 x 106 N/C. What is the magnitude of the torque on the
dipole in a uniform field when (a) the dipole is parallel to
the field, (b) the dipole is perpendicular to the field, and
(c) the dipole makes an angle of 30º to the field. 20º
14) An electron of charge, – e, and mass, m, and a positron of
charge, e, and mass, m, are in orbit around each other.
They are a distance, d, apart. The center of their orbit is
halfway between them. (a) Name the force that is acting
as the centripetal force making them move in a circle.
(b) Calculate the speed, v, of each charge in terms of e,
m, k (Coulomb’s Constant), and d.
15) A ball of mass, m, and positive charge, q, is dropped from
rest in a uniform electric field, E, that points downward.
If the ball falls through a height, h, and has a velocity of
gh2v , find its mass in terms of q, g, and E.
16)
The two charges above are fixed and cannot move. Find a
point in space where the total electric field will equal
zero.
ANSWERS:
1) a) 1.67 x 105 N/C,
WEST
b) 1.17 N, EAST
2) a) 7.97 x 105 N/C,
EAST
b) 5.6 N, EAST
3) 516 N/C, 61.3º
4) 717 N/C, 69.8º
5) a) 153.1 N/C,
SOUTH
b) 19.6 m/s2
c) 3.13 m/s
6) a) 6.71 x 1010 m/s2
b) 2.68 x 10–5 s
c) 24.1 m
d) 2.71 x 10–15 J
7) 1.23 x 106 N/C, 80.5º
8) 3.06 x 105 N/C, 48.4º
9) a) 3.04 m
b) 0.297 mm
10) 6.49 x 10–6 C
11) a) 222 ay
kqay4
b) 3y
kqa4
12) 5.81 N
13) a)
0
b) 7.2 N·m
c) 3.6 N·m
14)
md2
kev
15)
g
Eq
m
16) 8.2 cm
–
+
y q
a
a
–q
+
6 cm
–
– 4 μC 12 μC
d
– +
– q q 2q
+
Physics 226
Fall 2013
Problem Set #4
NOTE: Any answers of zero must have some kind of justification.
1) You have a thin straight wire of
charge and a solid sphere of charge.
The amount of charge on each object
is 8 mC and it is uniformly spread
over each object. The length of the
wire and the diameter of the sphere
are both 13 cm. (a) Find the amount
of charge on 3.5 cm of the wire.
(b) For the sphere, how much charge
is located within a radius of 3.5 cm
from its center?
2)
A uniform line of charge with density, λ, and length, L
is positioned so that its center is at the origin. See diagram
above. (a) Determine an equation (using integration)
for the magnitude of the total electric field at point P
a distance, d, away from the origin. (b) Calculate
the magnitude and direction of the electric field at P if
d = 2 m, L = 1 m, and λ = 5 μC/m. (c) Show that if
d >> L then you get an equation for the E-field that is
equivalent to what you would get for a point charge. (We
did this kind of thing in lecture.)
3)
A uniform line of charge with charge, Q, and length, L, is
positioned so that its center is at the left end of the line.
See diagram above. (a) Determine an equation (using
integration) for the magnitude of the x-component of the
total electric field at point P a distance, d, above the
left end of the line. (b) Calculate the magnitude and
direction of the x-component of the total electric field at
point P if d = 1.5 m, L = 2.5 m, and
Q
= – 8 μC.
(c) What happens to your equation from part (a) if d >>
L? Conceptually explain why this is true.
4)
13 cm
You have a semi-infinite line of charge with a uniform
linear density 8 μC/m. (a) Calculate the magnitude of
the total electric field a distance of 7 cm above the left
end of line. (You can use modified results from lecture
and this homework if you like … no integration
necessary.) (b) At what angle will this total E-field act?
(c) Explain why this angle doesn’t change as you move
far away from the wire. Can you wrap your brain around
why this would be so?
d
5)
A uniform line of charge with charge, Q, and length, D, is
positioned so that its center is directly below point P
which is a distance, d, above. See diagram above.
(a) Determine the magnitude of the x-component of the
total electric field at point P. You must explain your
answer or show calculations. (b) Calculate the magnitude
and direction of the y-component of the total electric field
at P if d = 2 m, D = 4.5 m, and Q = –12 μC. HINT: You
can use integration to do this OR you can use one of the
results (equations) we got in lecture and adapt it to this
problem.
6) You have an infinite line of charge of constant linear
density, λ. (a) Determine an equation for the magnitude
of the total electric field at point P a distance, d, away
from the origin. Use any method you wish (except Gauss’
Law) to determine the equation. There’s at least
three different ways you could approach this. You can
use the diagram in #5 where D → if you want a
visual. (b) Calculate the electric field at d = 4 cm with
λ = 3 μC/m.
P
+ + + + +
+
0 2
L
2
L
P
0
d
– – – – – – –
L
P
0
7 cm
+ + + + +
P
d
– – – – – – –
D
7)
You have three lines of charge each with a length of
50 cm. The uniform charge densities are shown. The
horizontal distance between the left plate and right ones is
120 cm. Find the magnitude and direction of the TOTAL
E-field at P which is in the middle of the left plate and the
right ones.
8) For the same charge distribution of problem #7, with the
exception that you change the sign of the 4 μC plate and
you change the distance between the plates to 160 cm,
find the magnitude and direction of the TOTAL E-field
at P which is in the middle of the left plate and the
right ones.
9)
You have 3 arcs of charge, two ¼ arcs and one ½ arc.
The arcs form of circle of radius 5 cm. The uniform linear
densities are shown in the diagram. (a) Using an integral
and showing your work, determine the equation for the
electric field at point P due to the ½ arc. (b) Calculate
the magnitude and direction of the total electric field at
point P.
10) For this problem use the same charge distribution as
problem #9, with the exception of changing all even
charges to the opposite sign. (a) Using an integral and
showing your work, determine the equation for the
electric field at point P due to the ½ arc. (b) Calculate
the magnitude and direction of the total electric field at
point P.
11) You have two thin discs both
of diameter 26 cm. They also
have the same magnitude surface
charge density of, 20 μC/m2, but
opposite sign. The charge is
uniformly distributed on the discs.
The discs are parallel to each
other and are separated by a distance of 30 cm.
(a) Calculate the magnitude and direction of the total
electric field at a point halfway between the discs along
their central axes. (b) Calculate the magnitude and
direction of the total electric field at a point halfway
between the discs along their central axes if the diameter
of the discs goes to infinity. (c) Determine the total
electric field at a point halfway between the discs along
their central axes if discs have charge of the same sign.
– 5 μC/m
+
+
+
+
+
+
–
–
–
3 μC/m
4 μC/m
P
12) You have two concentric thin rings of
charge. The outer ring has a dia-
meter of 50 cm with a uniformly
spread charge of – 15 μC. The inner
ring has a diameter of 22 cm with a
uniform linear charge density of
15 μC/m. Calculate the magnitude
and direction of the total E-field at
point P which lies 40 cm away from
the rings along their central axes.
P
13) A proton is released from rest 5 cm away from an infinite
disc with uniform surface charge density of 0.4 pC/m2.
(a) What is the acceleration of the proton once it’s
released? (b) Calculate the kinetic energy of the proton
after 2.5 s. [See Conversion Sheet for metric prefixes.]
2 μC/m
14)
In the above two diagrams, G & L, an electron is given
an initial velocity, vo, of 7.3 x 106 m/s above infinite
discs with uniform surface charge density of –0.15 fC/m2.
(a) In diagram G, how much time passes before the
electron stops? (b) In diagram L, how far does the
electron move horizontally after it has traveled 20 m
vertically? (Hint: Think projectile motion)
15) Two thin infinite planes
of surface charge density
6 nC/cm2 intersect at 45º
to each other. See the
diagram in which the
planes are coming out of
the page (edge on view).
Point P lies 15 cm from
each plane. Calculate
the magnitude and
direction of the total
electric field at P.
–
+
+ +
– –
+
– – 2 μC/m 5 μC/m
+
+
+
P
– –
L G
vo
vo
P
45º
P
– +
ANSWERS:
1) a) 2.15 mC
b) 1.25 mC
2) a) 22 Ld4
Lk4
b) 1.2 x 104 N/C,
EAST
3) a)
22x Ld
d
1
dL
Qk
E
b) 9322 N/C, EAST
c) 0
4) a) 1.46 x 106 N/C
b) 45º
c) Because Ex = Ey
5) a) 0
b) 1.79 x 104 N/C, SOUTH
6) 1.35 x 106 N/C, NORTH
7) 5.93 x 104 N/C, 13.6º
8) 2.37 x 104 N/C, 59.8º
9) a)
R
k2Ey
b) 4.85 x 105 N/C, 22.0º
10) a)
R
k2E y
b) 2.05 x 106 N/C, 74.8º
11) a) 5.53 x 105 N/C, WEST
b) 2.26 x 106 N/C, WEST
c) 0
12) 1.01 x 105 N/C, WEST
13) a) 2.17 x 106 m/s2
b) 2.45 x 10–14 J
14) a) 4.9 s
b) 3780 m
15) 2.6 x 106 N/C, 22.5º
Physics 226
Fall 2013
Problem Set #5
NOTE: Any answers of zero must have some kind of justification.
1)
A uniform electric field of strength 300 N/C at an angle of
30º with respect to the x-axis goes through a cube of sides
5 cm. (a) Calculate the flux through each cube face:
Front, Back, Left, Right, Top, and Bottom. (b) Calculate
the net flux through the entire surface. (c) An electron is
placed centered 10 cm from the left surface. What is the
net flux through the entire surface? Explain your answer.
2)
A right circular cone of height 25 cm and radius 10 cm is
enclosing an electron, centered 12 cm up from the base.
See Figure G. (a) Using integration and showing all work,
find the net flux through the cone’s surface. The electron
is now centered in the base of the cone. See Figure L. (b)
Calculate the net flux through the surface of the cone.
3) Using the cube in #1, you place a 4μC charge directly in
the center of the cube. What is the flux through the top
face? (Hint: Consider that this problem would be MUCH
more difficult if the charge was not centered in the cube.)
4) Using the cube in #1, you place a 4μC charge at the lower,
left, front corner. What is the net flux through the cube?
(Hint: Think symmetry.)
5) You have a thin spherical shell
of radius 10 cm with a uni-
form surface charge density of
– 42 μC/m2. Centered inside the
sphere is a point charge of 4 μC.
Find the magnitude and direction
of the total electric field at:
(a) r = 6 cm and (b) r = 12 cm.
6) You have a solid sphere of
radius 6 cm and uniform
volume charge density of
– 6 mC/m3. Enclosing this is
a thin spherical shell of
radius 10 cm with a total
charge of 7 μC that is
uniformly spread over the
surface. (a) What is the
discontinuity of the E-field at
the surface of the shell. (b) What is the discontinuity of
the E-field at the surface of the solid sphere? Also, find
the magnitude and direction of the total electric field at:
(c) r = 4 cm, (d) r = 8 cm, and (e) r = 13 cm.
x
30º
y
7) Use the same set-up in #6 with the following exceptions:
The solid sphere has a total charge of 5 μC and the shell
has uniform surface charge density of 60 μC/m2. Answer
the same questions in #6, (a) – (e).
8) You have a thin infinite
cylindrical shell of radius 8 cm
and a uniform surface charge
density of – 12 μC/m2. Inside the
shell is an infinite wire with a
linear charge density of 15 μC/m.
The wire is running along the
central axis of the cylinder.
(a) What is the discontinuity of the E-field at the surface
of the shell? Also, find the magnitude and direction of the
total electric field at: (b) r = 4 cm, and (c) r = 13 cm.
9) You have a thin infinite
cylindrical shell of radius 15 cm
and a uniform surface charge
density of 10 μC/m2. Inside the
shell is an infinite solid cylinder
of radius 5 cm with a volume
charge density of 95 μC/m3.
The solid cylinder is running
along the central axis of the
cylindrical shell. (a) What is the discontinuity of the
E-field at the surface of the shell? (b) What is the
discontinuity of the E-field at the surface of the solid
cylinder. Also, find the magnitude and direction of the
total electric field at: (c) r = 4 cm, (d) r = 11 cm, and
(e) r = 20 cm.
–
–
G L
+
10) You have a thick spherical shell
of outer diameter 20 cm and
inner diameter 12 cm. The shell
has a total charge of – 28 μC
spread uniformly throughout the
object. Find the magnitude and
direction of the total electric field
at: (a) r = 6 cm, (b) r = 15 cm,
and (c) r = 24 cm.
11) You have a thick cylindrical shell
of outer diameter 20 cm and
inner diameter 12 cm. The shell
has a uniform volume charge
density of 180 μC/m3. Find
the magnitude and direction
of the total electric field at:
(a) r = 6 cm, (b) r = 15 cm, and
(c) r = 24 cm.
12)
You have an thin infinite sheet of charge with surface
charge density of 8 μC/m2. Parallel to this you have a
slab of charge that is 3 cm thick and has a volume charge
density of – 40 μC/m3. Find that magnitude and
direction of the total electric field at: (a) point A which
is 2.5 cm to the left of the sheet, (b) point B which is
4.5 cm to the right of the sheet, and (c) point C which is
1 cm to the left of the right edge of the slab.
13)
You have an infinite slab of charge that is 5 cm thick and
has a volume charge density of 700 μC/m3. 10 cm to
the right of this is a point charge of – 6 μC. Find that
magnitude and direction of the total electric field at:
(a) point A which is 2.5 cm to the left of the right edge of
the slab, (b) point B which is 6 cm to the right of the
slab, and (c) point C which is 4 cm to the right of the
point charge.
14) You have two infinite sheets of charge
with equal surface charge magnitudes
of 11 μC/m2 but opposite signs. Find
the magnitude and direction of the
total electric field, (a) to the right of
the sheets, (b) in between the sheets,
and (c) to the left of the sheets.
15)
A hydrogen molecule (diatomic hydrogen) can be
modeled incredibly accurately by placing two protons
(each with charge +e) inside a spherical volume charge
density which represents the “electron cloud” around the
nuclei. Assume the “cloud” has a radius, R, and a net
charge of –2e (one electron from each hydrogen atom)
and is uniformly spread throughout the volume. Assume
that the two protons are equidistant from the center of
the sphere a distance, d. Calculate, d, so that the protons
each have a net force of zero. The result is darn close to
the real thing. [This is actually a lot easier than you
think. Start with a Free-Body Diagram on one proton
and then do F = ma.]
ANSWERS:
NOTE: Units for 1 – 4
are CmN 2
1) a) 0 for F/B,
0.375 for L/R,
0.65 for T/B
b) & c) 0
2) a) – 1.81 x 10–8
b) – 9.05 x 10–9
3) 7.54 x 104
4) 5.66 x 104
5) a) 9.99 x 106 N/C,
OUTWARD [O]
b) 7.99 x 105 N/C
INWARD [I]
6) a) 6.29 x 106 N/C
b) 0
c) 9.04 x 106 N/C, I
d) 7.63 x 106 N/C, I
e) 8.36 x 105 N/C, O
7) a) 6.78 x 106 N/C
b) 0
c) 4.99 x 105 N/C, O
d) 7.03 x 106 N/C, O
e) 6.67 x 106 N/C, O
8) a) 1.36 x 106 N/C
b) 6.74 x 106 N/C, O
c) 1.24 x 106 N/C, O
9) a) 1.13 x 106 N/C
b) 0
c) 2.15 x 105 N/C, O
d) 1.22 x 105 N/C, O
e) 9.15 x 105 N/C, O
10) a) 0
b) 2.94 x 106 N/C, I
c) 4.37 x 106 N/C, I
11) a) 0
b) 5.49 x 105 N/C, O
c) 1.09 x 106 N/C, O
12) a) 3.84 x 105 N/C, L
b) 5.20 x 105 N/C,
R
c) 4.30 x 105 N/C, R
13) a) 3.84 x 105 N/C, R
b) 3.57 x 107 N/C, R
c) 3.18 x 105 N/C, L
14) a) 0
b) 1.24 x 106 N/C, R
c) 0
15) 0.794R
10 cm
A B C
10 cm
A B
–
C
R
+ +
d d
Physics 226
Fall 2013
Problem Set #6
NOTE: Any answers of zero must have some kind of justification.
1) You have a cylindrical metal shell of
inner radius 6 cm and outer radius
9 cm. The shell has no net charge.
Inside the shell is a line of charge of
linear density of – 7 μC/m. Find the
magnitude and direction of the electric
field at (a) r = 3 cm, (b) r = 7 cm,
and (c) r = 13 cm. Also, calculate the surface charge
density of the shell on (d) the inner surface and (e) the
outer surface.
2) You have a uniformly charged
sphere of radius 5 cm and
volume charge density of
– 7 mC/m3. It is surrounded
by a metal spherical shell
with inner radius of 10 cm
and outer radius of 15 cm.
The shell has a net charge
8 μC. (a) Calculate the total
charge on the sphere. Find
the magnitude and direction of the electric field at
(b) r = 13 cm and (c) r = 18 cm. Also, calculate the
surface charge density of the shell on (d) the inner surface
and (e) the outer surface.
3)
Two 2 cm thick infinite slabs of metal are positioned as
shown in the diagram. Slab B has no net charge but Slab
A has an excess charge of 5 μC for each square meter. The
infinite plane at the origin has a surface charge density of
– 8 μC/m2. Find the magnitude and direction of the
electric field at (a) x = 2 cm, and (b) x = 4 cm. Also,
calculate the surface charge density on (c) the left edge of
A, (d) the right edge of A, and (e) the left edge of B.
4)
A positive charge of 16 nC is nailed down with a #6 brad.
Point M is located 7 mm away from the charge and point
G is 18 mm away. (a) Calculate the electric potential at
Point M. (b) If you put a proton at point M, what
electric potential energy does it have? (c) You release the
proton from rest and it moves to Point G. Through what
potential difference does it move? (d) Determine the
velocity of the proton at point G.
5)
All the charges above are multiples of “q” where
q = 1μC. The horizontal and vertical distances between
the charges are 25 cm. Find the magnitude of the net
electric potential at point P.
6) Use the same charge distribution as in problem #5 but
change all odd-multiple charges to the opposite sign. Find
the magnitude of the net electric potential at point P.
7) A parallel plate setup has a distance
between the plates of 5 cm. An
electron is place very near the negative
plate and released from rest. By the
time it reaches the positive plate it has
a velocity of 8 x 106 m/s. (a) As the
electron moves between the plates what
is the net work done on the charge? (b) What is the
potential difference that the electron moves through?
(c) What is the magnitude and direction of the electric
field in between the plates?
3 cm 5 cm 8 cm 0 10 cm
A B
– 8q
– 4q
+9q
+9q
– 5q
+6q +6q
+2q
P
+
M
G
–
8)
A uniform line of charge with density, λ, and length, L is
positioned so that its left end is at the origin. See diagram
above. (a) Determine an equation (using integration) for
the magnitude of the total electric potential at point P a
distance, d, away from the origin. (b) Calculate the
magnitude of the electric potential at P if d = 2 m,
L = 1 m, and λ = – 5 μC/m. c) Using the equation
you derived in part a), calculate the equation for the
electric field at point P. It should agree with the result
we got in Lecture Example #19.
9) You have a thin spherical shell
of radius 10 cm with a uni-
form surface charge density of
11 μC/m2. Centered inside the
sphere is a point charge of
– 4 μC. Using integration, find
the magnitude of the total electric
potential at: (a) r = 16 cm and
(b) r = 7 cm.
10) You have a uniformly
charged sphere of radius
5 cm and volume charge
density of 6 mC/m3. It is
surrounded by a metal
spherical shell with inner
radius of 10 cm and outer
radius of 15 cm. The
shell has no net charge.
Find the magnitude of the
electric potential at (a) r = 20 cm, (b) r = 12 cm, and
(c) r = 8 cm.
11) Use the same physical situation with the exception
of changing the inner sphere to a solid metal with
a surface charge density of 9 μC/m2 and giving the
shell a net charge of – 3 μC. Find magnitude of
the electric potential at (a) r = 20 cm, (b) r = 12 cm,
(c) r = 8 cm, and (d) r = 2 cm.
12) CSUF Staff Physicist & Sauvé Dude, Steve
Mahrley, designs a lab experiment that
consists of a vertical rod with a fixed bead of
charge Q = 1.25 x 10–6 C at the bottom. See
diagram. Another bead that is free to slide on
the rod without friction has a mass of 25 g and
charge, q. Steve releases the movable bead
from rest 95 cm above the fixed bead and it
gets no closer than 12 cm to the fixed bead.
(a) Calculate the charge, q, on the movable
bead. Steve then pushes the movable bead
down to 8 cm above Q. He releases it from rest.
(b) What is the maximum height that the bead reaches?
13)
d
P
0
– – – – –
+
L 20 cm
You have two metal spheres each of diameter 30 cm that
are space 20 cm apart. One sphere has a net charge of
15 μC and the other – 15 μC. A proton is placed very
close to the surface of the positive sphere and is release
from rest. With what speed does it hit the other sphere?
14) A thin spherical shell of radius, R, is centered at the
origin. It has a surface charge density of 2.6 C/m2.
A point in space is a distance, r, from the origin. The
point in space has an electric potential of 200 V and an
electric field strength of 150 V/m, both because of the
sphere. (a) Explain why it is impossible for r < R.
(b) Determine the radius, R, of the sphere.
– 4 μC 12 μC 15) – – +
6 cm
The two charges above are fixed and cannot move. Find a
point in space where the total electric potential will equal
zero.
ANSWERS:
1) a) 4.20 x 106 N/C, I
b) 0
c) 9.68 x 105 N/C, I
d) 1.86 x 10–5 C/m2
e) – 1.24 x 10–5 C/m2
2) a) – 3.67 x 10–6 C
b) 0
c) 1.20 x 106 N/C, O
d) 2.92 x 10–5 C/m2
e) 1.73 x 10–6 C/m2
3) a) 7.35 x 105 N/C, L
b) 0
c) 6.5 x 10–6 C/m2
d) – 1.5 x 10–6 C/m2
e) 1.5 x 10–6 C/m2
f) – 1.5 x 10–6 C/m2
4) a) 2.06 x 104 V
b) 3.29 x 10–15 J
c) – 1.26 x 104 V
d) 4.91 x 105 m/s
5) 5.02 x 105 V
6) – 7.87 x 104 V 7)
a) 2.92 x 10-17 J
b) 182.2 V
c) 3644 N/C
8) a)
d
Ldlnk
b) – 1.83 x 104 V
9) a) – 1.47 x 105 V
b) – 3.90 x 105 V
10) a) 1.41 x 105 V
b) 1.88 x 105 V
c) 2.59 x 105 V
11) a) – 8.37 x 104 V
b) – 1.12 x 105 V
c) – 8.62 x 104 V
d) – 9900 V
12) a) 2.48 x 10–6 C
b) 1.42 m
13) 1.4 x 107 m/s
14) 2.86 m
15) 1.5 cm
q
Q
Physics 226
Fall 2013
Problem Set #7
1) You have a parallel plate capacitor of plate separation
0.1 mm that is filled with a dielectric of neoprene rubber.
The area of each plate is 1.8 cm2. (a) Calculate the
capacitance of the capacitor. The capacitor is charged by
taking electrons from one plate and depositing them on
the other plate. You repeat this process until the potential
difference between the plates is 350 V. (b) How many
electrons have been transferred in order to accomplish
this?
2) A capacitor with ruby mica has an effective electric field
between the plates of 4600 V/m. The plates of the
capacitor are separated by a distance of 4 mm. 50 mJ of
energy is stored in the electric field. (a) What is the
capacitance of the capacitor? (b) Calculate the energy
density in between the plates.
3) A capacitor with a dielectric of paper is charged to 0.5 mC.
The plates of the capacitor are separated by a distance of
8 mm. 40 mJ of energy is stored in the electric field.
(a) What is the strength of the effective electric field?
(b) Calculate the energy density in between the plates.
4) A capacitor of 10 μF is charged by connecting it to a
battery of 20 V. The battery is removed and you pull the
plates apart so that you triple the distance between them.
How much work do you do to pull the plates apart?
5) The flash on a disposable camera contains a capacitor
of 65 F. The capacitor has a charge of 0.6 m C stored on
it. (a) Determine the energy that is used to produce a
flash of light. (b) Assuming that the flash lasts for 6 ms,
find the power of the flash. (Think back to 225.)
6) A spherical shell conductor of
radius B encloses another spherical
shell conductor of radius A. They
are charged with opposites signs
but same magnitude, q. (a) Using
integration, derive an equation for
the capacitance of this spherical
capacitor. (b) Calculate the
capacitance if A = 45 mm and
B = 50 mm. (c) If q = 40 μC, what is the energy density
in between the shells?
7) You attach a battery of 15 V to an air-filled capacitor of
5 μF and let it charge up. (a) If the plate separation is
3 mm, what is the energy density in between the plates?
You then remove the battery and attach the capacitor to a
different uncharged capacitor of 2 μF. (b) What is the
amount of charge on each capacitor after they come to
equilibrium?
8) You attach a 100 pF capacitor to a battery of 10 V. You
attach a 250 pF battery to 7 V. You remove both of the
batteries and attach the positive plate of one capacitor to
the positive plate of the other. After they come to
equilibrium, find the potential difference across each
capacitor.
9) Do problem #8 but when you attach the capacitors
together attach the opposite sign plates instead of the
same sign plates.
10)
Determine the equivalent capacitance between points A
and B for the capacitors shown in the circuit above.
11)
Determine the equivalent capacitance between points A
and B for the capacitors shown in the circuit above.
12) Design a circuit that has an equivalent capacitance of
1.50 μF using at least one of each of the follow
capacitors: a 1 μF, a 2 μF, and a 6 μF. [You must also
show where your A and B terminals are located.]
A
20 F
4 F
4 F
6 F
12 F
B
30 F
A
12 F
18 F
6 F
20 F
B
12 F 75 F
13)
The two capacitors above both have plates that are
squares of sides 3 cm. The plate separation is 1.2 cm for
both. Between each of the capacitor plates are two
different dielectrics of neoprene rubber and Bakelite.
Everything is drawn to scale. Find the capacitance of
each capacitor. (HINT: Think series and parallel.)
14) The plates of an air-filled capacitor have area, A, and are
separated by a distance, d. The capacitor is charged by a
battery of voltage, V. Three things are going to change:
(1) The plates of the capacitor are pulled apart so that
the distance between the plates triples. (2) The area of
the plates increase by a factor of 6. (3) The voltage of
the battery decreases by a factor of 4. Determine
expressions in terms of A, d, and/or V for (a) the new
capacitance, (b) the new charge, and (c) the new energy
density.
15)
A massless bar of length, L, is hanging from a string that
is attached 1/3 of the length of the bar from the right
end. A block of mass, M, is hung from the right end.
The left end of the bar has an air-filled massless capacitor
of plate area, A, and plate separation, d. Find an
expression for the potential difference between the plates
so that this system is in equilibrium. (HINT: You will
need the equation
dx
dU
F from 225.)
ANSWERS:
(a) (b)
1) a) 1.067 x 10–10 F
b) 2.34 x 1011 e–
2) a) 2.95 x 10–4 F
b) 5.05 x 10–4 J/m3
3) a) 2 x 104 V/m
b) 6.7 x 10–3 J/m3
4) 4 x 10–3 J
5) a) 2.8 x 10–3 J
b) 0.467 W
6) a)
AB
AB4C o
b) 5.01 x 10–11 F
c) 1.125 x 105 J/m3
7) a) 1.11 x 10–4 J/m3
b) 2.14 x 10–5 C,
5.36 x 10–5 C
8) 7.86 V
9) 2.14 V
10) 4 μF
11) 9 μF
13) a) 3.85 pF
b) 3.76 pF
14) a)
d
A2
C o
b)
d2
AV
Q o
c) 2
2
o
d288
Vu
15)
A
Mg
dV
o
M
Physics 226
Fall 2013
Problem Set #8
1) Analyze the circuit below using a QCV chart. You must
show appropriate work for full credit.
2) Analyze the circuit below using a QCV chart. You must
show appropriate work for full credit.
3) Analyze the circuit below using a QCV chart. You must
show appropriate work for full credit.
4)
An Oppo Digital Blu-Ray player [DMP-95] (Yes, I am an
audiophile.) has a power cable which has a metal that
allows 9 x 1019 electrons per cubic millimeter. On average,
the cable passes 1 x 1022 electrons every hour. The
electrons passing through the player have a drift velocity
of 4.5 μm/s. (a) What current does the Oppo draw?
(b) Calculate the diameter of the cable?
5) The Large Hadron
Collider at CERN creates
proton beams which
collide together resulting
in pictures like the one
at the right. Some of
these beams can have a
radius of 1.1 mm with a
current of 1.5 mA. The
kinetic energy of each
proton in this beam is 2.5 MeV. (a) Calculate the
number density of the protons in the beam. (b) If the
beam is aimed at a metal target, how many protons would
strike the screen in 1 minute?
C1 = 8 μF C2 = 15 μF
20 V
C3 = 30 μF
6)
Two copper wires are soldered together. Wire #1 has a
radius of 0.7 mm. Wire #2 has a radius of 1.2 mm.
Copper has a number density of 8.47 x 1028 e–/m3. The
drift velocity in Wire #1 is 0.72 mm/s. If you want the
current to remain the same in both, what is the drift
velocity in Wire #2?
7) A nichrome cable has a current of 140 A running through
it. Between two points on the cable that are 0.22 m apart,
there is a potential difference of 0.036 V (a) Calculate the
diameter of the cable. (b) How much heat energy does
this part of the wire emit in 1 minute?
8) A “Rockstar” toaster uses a
tungsten heating element
(wire). The wire has a
diameter of 1.2 mm. When
the toaster is turned on at
20 C, the initial current is
1.6 A. (a) What is the
current density in the wire?
(b) A few seconds later,
the toaster heats up and the current is 1.20 A. What is
the temperature of the wire? (c) If the toaster is plugged
into a standard wall outlet in Kankakee, Illinois, what is
the rate that energy is dissipated from the heating
element?
9) Skid runs a 10 mile line of copper cable out to his shack in
the sticks so he can have electricity to play Lord of the
Rings Online. At 20ºC the resistance of the cable is 12 .
At 50ºC the cable emits 1.5 kJ every second. (a) What is
the resistance of the cable at 50ºC? (b) What is the
current running through the cable at 50ºC? (c) Calculate
the current density at 50ºC.
C1 = 18 F
Wire #1 Wire #2
C2 = 6 μF
C3 = 4 μF
C4 = 30 μF 25 V
C1 = 5 F
C2 = 4 μF
15 V
C3 = 1 μF
C4 = 12 μF
10) A modern hair dryer uses a
nichrome heating element
that typically is 30-gauge
wire around 40 cm in length.
The gauge rating on a wire
refers to its diameter. In this
case, 30-gauge wire has a
diameter of 0.254 mm.
Nichrome has a number
density of 7.94 x 1028 e–/m3.
If the drift velocity of the electrons in the wire is
18.7 mm/s, what is the voltage that the hair dryer is
plugged into?
11) Before LCD, LED, Plasma,
and (the latest) OLED
TVs, there were CRT
(Cathod-Ray Tube) TVs.
Inside these TVs were
electron guns that shot an
electron beam of diameter
0.5 mm and current
density of 244 A/m2 onto
the inside of a glass screen
which was coated with phosphor. How many electrons
would hit the phosphor every minute?
12)
Determine the equivalent resistance between points A
and B for the resistors shown in the circuit above.
13)
Determine the equivalent resistance between points A
and B for the resistors shown in the circuit above.
14)
Determine the equivalent resistance between points A
and B for the resistors shown in the circuit above.
15) Design a circuit that has an equivalent resistance of
1.00 using at least one of each of the follow resistors:
a 1 , a 2 , and a 6 . [You must also show where your
A and B terminals are located.]
ANSWERS:
NOTE: Some of these answers are minimal since there are
checks that you can do to verify your answers.
A
2
7
B
5
4
8
30
1
6
14
10
30
B
18
96
6
32 18
60 A
A
20
30
B
30
7
50
12
4
5
60
1) CEQ = 18 μF 8) a) 1.415 x 106 A/m2
2) CEQ = 6 μF b) 94.1ºC
3) CEQ = 2 μF c) 144 W
9) a) 13.4 4) a) 0.444 A
b) 2.96 mm b) 10.58 A
c) 5.14 x 105 A/m2 5) a) 1.13 x 1014 p+/m3
b) 5.63 x 1017 p+ 10) 95.0 V
6) 0.262 mm/s 11) 1.8 x 1016 e–
7) a) 0.033 m 12) 4
b) 302 J 13) 14
14) 22
Physics 226
Fall 2013
Problem Set #9
NOTE: You can only use circuit tricks on 9 – 11 but not on any others.
1) Analyze the following circuit using a VIR chart.
2) Swap the location of the battery and R1 in the circuit from
problem #1. Analyze the circuit using a VIR chart.
3) Analyze the following circuit using a VIR chart.
4) The battery in this problem has an internal resistance of
0.15 . (a) Analyze the following circuit using a VIR
chart. (b) Is this circuit well designed? Discuss, explain.
5) Analyze the following circuit using a VIR chart.
6) Analyze the following circuit using a VIR chart.
7) The battery in this problem has an internal resistance of
1 . (a) Analyze the following circuit using a VIR chart.
(b) Is this circuit well designed? Discuss, explain.
8) A load of 3.5 is connected across a 12 V battery. You
measure a voltage of 9.5 V across the terminals of the
battery. (a) Find the internal resistance of the battery.
(b) Is this circuit well designed? Discuss, explain.
9) Analyze the circuit from problem
#5 using a VIR chart. You are
using only the diagram in #5, not
the values. New values are given at
the right. You may use a circuit
trick for this circuit, but only for
ONE value.
10) Analyze the circuit from problem
#6 using a VIR chart. You are
using only the diagram in #6, not
the values. New values are given
at the right. You may use a circuit
trick for this circuit, but only for
ONE value.
R1
20 V
R2
R3
R4
R5
Given:
R1 = 12
R2 =
3
R3 = 8
R4 = 36
R5 = 15
50 V
R1 Given:
R1 = 28
R2 = 6
R3 = 84
R4 = 7
R5 = 54
R3
R2
R4
R5
55 V
R1 Given:
R1 = 18
R2 = 32
R3 = 15
R4 = 2
1
R5 = 42
R6 = 30
R7 = 52
R3
R2
R4 R5
R6 R7
R1
VB
R2
R3 R4
Given:
VB = 60 V
V2 = 50 V
I1 = 2 A
I4 = 3 A
R3 = 8
R1
VB
R2 R3
R4
R5
Given:
V5 = 32 V
I2 = 0.4 A
I4 = 0.5 A
R1 = 36 R6
R3 = 60
R4 = 36
R6 = 32
R1
VB
Given:
VB = 32 V
R2 I1 = 4 A R3
R3 = 12
R4 R4 = 8
Given:
VB = 63 V
R1 = 8
R2 = 20
R3 = 35
R4 = 4
9
Given:
VB = 75 V
R1 = 16
R2 = 40
R3 = 48
R4 = 24
R5 = 8
R6 = 24
11) Analyze the following circuit using a VIR chart.
12) Using the information you are
given for the circuit at the
right, answer the following.
(a) Determine the magnitude
and direction of the current in
the circuit. (b) Determine
which point, A or B, is at a
higher potential.
13) Calculate the unknown currents I1, I2, and I3 for the circuit
below.
14) Calculate the unknown currents I1, I2, and I3 for the circuit
below.
Given:
15) Calculate the unknown currents I1, I2, and I3 for the circuit
below.
ANSWERS:
NOTE: These answers are minimal since there are checks
that you can do to verify your answers.
R1 R2
R3 R4
R5 R6
I1 8 V VB = 50 V
R1 = 9
R2 = 4
R3 = 18
R4 = 4
R5 = 7
R6 = 12
B
A
17 V
13 7
5
11
23 V
6
1
10 V
25 V
3
5
7
I1
I2
I3
4
9
10
4 7
I2
6
I3 22 V
3 10 V I1
4
4 25 V
2 5
I2
I3
20 V 4
7) REQ = 8 1) REQ = 2
8) a) 0.923 2) REQ = 11.48
9) REQ = 21 3) REQ = 25
10) REQ = 25 4) REQ = 12.15
11) REQ = 20 5) REQ = 12
12) a) 1.11 A 6) REQ = 40
