226_problem_sets_-_e1

Physics226

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 let’s 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?

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?

Mitch Sofa Skid

y

F2 = 90 N

F1 = 40 N 35

8) Derivatives:

a) Given: Lx2Lx4y 22 , 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

kx , 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) 4.5s m/s

5) 103.3 m

6) 78.8 N

7) 1.78N

8) a) 24×2

+

8xL

–

4

L

b) 22 x4L

L4

9) a)

r

k2

b)

22 xR

x1k2

45 x

F3 = 60 N

y

F2 = 65 N

F1 = 45 N 60

50 x

70

F3 = 60 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

4 μC 12 μC

+ +

8 cm

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

15)

+6q – 4q

+4q

+3q +3q

– 1q

+8q

m

12 cm

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 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, →

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

6) FE = 24.3 N, → 15) 10.56 kg

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 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

travels 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

S M

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)

An 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 and 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 the magnitude of the torque on the

dipole torque on 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) 6.08 x 104 N/C,

WEST

b) 0.426 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) 2.97 m

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

– 4 μC 12 μC

– +

– +

– q

d

q q

+

6 cm

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)

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

13 cm

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 buy 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

– 4 μC/m

–

+

+

+

P

3 μC/m

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.

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

– +

P

P

– –

– –

L

vo

vo

G

P

45º

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

d1

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) 3.63 x 105 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

R

k210) a) E 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.

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.

x

30º

y

–

–

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)

R

+ +

d d

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 that you

think. Start with a Free-Body Diagram on one proton

and then do F = ma.]

10 cm

B C A

ANSWERS:

1) a) 0, 0.375 Wb,

00.65 Wb, Resp

b) & c) 0

2) a) – 1.81 x 10–8 Wb

b) – 9.05 x 10–9 Wb

3) 7.54 x 104 Wb

4) 5.66 x 104 Wb

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) 9.04 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) 5.30 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

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 the 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 magnitude and the direction of the

electric field at (a) x = – 2 cm, (b) x = 2 cm, (c) x = 4 cm,

(d) x = 7 cm, and (e) x = 12 cm. Also, calculate the

surface charge density on (e) the left edge of A, (f) the

right edge of A, (g) the left edge of B, and (f) the right

edge of B.

4)

A positive charge of 16 μC 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 even-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 7.97 x 105 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?

– 8q

– 4q

A B

0 3 cm 5 cm 8 cm 10 cm

+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 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:

6) – 7.87 x 104 V 7)

a) 2.92 x 10-17 J

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) 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

q 10) a) 1.41 x 105 V

b) 1.88 x 105 V

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

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 Q

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