beldorin (sb37529) – Homework 8 Sources of Magnetism – catala – (20491)

This print-out should have 23 questions.

Multiple-choice questions may continue on

the next column or page – ﬁnd all choices

before answering.

001 (part 1 of 5) 10.0 points

A solid nonconducting cylinder of radius R

is very long. It contains a negative charge

evenly distributed throughout the cylinder,

with volume charge density ρ. Point P1 is

outside the cylinder at a distance r1 from its

center C and point P2 is inside the cylinder at

a distance r2 from its center C . Both points

are in the same plane, which is perpendicular

to the axis of the cylinder.

P1

3.

002 (part 2 of 5) 10.0 points

Using Gauss’s law, derive expressions for the

magnitude of the electric ﬁeld E in terms of

r , R, ρ, and fundamental constants for the

following case r > R (outside the cylinder).

1. E =

2. E =

r1

R

P2

On the cross-sectional diagram containing

P1 and P2 , select the vector diagram to indicate the directions of the electric ﬁeld at

points P1 and P2 .

P1

3. E =

4. E =

5. E =

6. E =

7. E =

8. E =

1.

C

C

P2

P1

C

r2

1

ρR

2

0r

ρ R2

2 0r

ρ R2

0r

ρR

2 0r

ρR

0r

ρ R2

2 0 r2

ρR

2 0 r2

ρ R2

2

0r

003 (part 3 of 5) 10.0 points

Using Gauss’s law, derive expressions for the

magnitude of the electric ﬁeld E in terms of

r , R, ρ, and fundamental constants for the

following case r < R (inside the cylinder).

P2

P1

1. E =

2.

C

2. E =

P2

3. E =

4. E =

5. E =

ρ

2 r2 0

ρ r2

20

ρ

2

r0

ρ

r0

ρr

20

beldorin (sb37529) – Homework 8 Sources of Magnetism – catala – (20491)

6. E =

ρ r2

2

P1

0

7. E =

8. E =

ρ

2r

ρr

3.

0

P2

0

004 (part 4 of 5) 10.0 points

Another cylinder of the same dimensions, but

made of conducting material, carries a total

current I parallel to the length of the cylinder, as shown in the diagram above. The current density is uniform throughout the crosssectional area of the cylinder. Points P1 and

P2 are in the same positions with respect to

the cylinder as they were for the nonconducting cylinder.

005 (part 5 of 5) 10.0 points

Use Ampere’s law to derive an expression for

the magnetic ﬁeld B inside the cylinder in

terms of r , R, I , and fundamental constants.

1. B =

2. B =

P1

3. B =

r1

I

R

C

I

r2

P2

On the cross-sectional diagram containing

P1 and P2 in which the current is out of the

plane of the page (toward the reader), select

the vector diagram to indicate the directions

of the magnetic ﬁeld at points P1 and P2 .

P1

1.

C

P2

P1

2.

C

P2

C

4. B =

5. B =

6. B =

7. B =

8. B =

µ0 I

π

µ0 I

π

µ0 I

2π

µ0 I

2π

µ0 I

2π

µ0 I

π

µ0 I

π

µ0 I

2π

·

·

·

·

·

·

·

·

r2

R2

r

R

r2

R2

r

R2

r2

R

r

R2

r2

R

r

R

006 10.0 points

A coil of 228 turns carrying a current of

150 mA is wound on a ring with a rectangular cross section of inner radius 2.2 cm,

outer radius 8.9 cm, and height 2.2 cm.

If the ring has a magnetic permeability

µ = 2000 µ0 , what is the ﬂux through a cross

section of the ring?

Answer in units of µWb.

007 10.0 points

Two parallel copper rods are 0.9 cm apart.

Lightning sends a 10 kA pulse of current along

each conductor.

The permeability of free space is 1.25664 ×

10−6 N/A2 .

Calculate the force per unit length on one

conductor.

beldorin (sb37529) – Homework 8 Sources of Magnetism – catala – (20491)

Answer in units of N/m.

008 10.0 points

Gauss’ law for magnetism tells us

1. the magnetic ﬁeld of a current element.

2. the net charge in any given volume.

3. that the line integral of a magnetic ﬁeld

around any closed loop vanishes.

4. that the magnetic monopoles do not exist.

5. that charges must be moving to produce

magnetic ﬁelds.

009 10.0 points

Consider two parallel wires where the magnitude of the left current is 2 I0 and that of

the right current is I0 . Point A is midway between the wires, and B is an equal distance

on the other side of the wires.

A

2.

3.

4.

5.

6.

BA

BB

BA

BB

BA

BB

BA

BB

BA

BB

BA

BB

=

2

3

=4

5

2

4

=

3

=

1

3

1

=

2

=

=9

P

2A×

B

0. 8 m

x

2A

C

Which of the diagrams correctly denotes

the directions of the components of the

magnetic ﬁeld from each conductor at the

point P ? The permeability of free space is

4 π × 10−7 N/A2 .

=3

=0

=2

010 (part 1 of 4) 10.0 points

Four long, parallel conductors carry equal 2 A

currents. A cross-sectional view of the conductors is shown in the ﬁgure. Each side of

the square has length of 0.8 m. The current

direction is out of the page at points indicated by the dots and into the page at points

indicated by the crosses.

y

A

D

×

2A

2A

B

The ratio of the magnitude of the magnetic

ﬁeld at point A to that at point B is

1.

BA

BB

B

8. A

BB

B

9. A

BB

B

10. A

BB

7.

3

BB

BC

1.

P

BA

BD

beldorin (sb37529) – Homework 8 Sources of Magnetism – catala – (20491)

BD

013 (part 4 of 4) 10.0 points

What is the direction of this resultant magnetic ﬁeld?

BA

BB

2.

P

BC

1.

2.

BD

3.

3.

P

BC

BB

4.

BA

5.

6.

4.

P

BC

BB

BA

7.

BD

8.

BA

5.

9.

P

BC

BB

BD

BA

BB

6.

4

P

BC

BD

011 (part 2 of 4) 10.0 points

What is the magnitude of each of the four

components BA , BB , BC , and BD at the point

P?

Answer in units of T.

012 (part 3 of 4) 10.0 points

What is the magnitude of the magnetic ﬁeld

at point P ?

Answer in units of T.

Bnet

Bnet

Bnet

Bnet

Bnet

Bnet

Bnet

Bnet

Bnet

Bnet

Bnet

Bnet

Bnet

Bnet

Bnet

Bnet

Bnet

Bnet

1

ı

= √ (−ˆ − ˆ)

2

is undetermined, since B = 0

= −ˆ

ı

1

ı

= √ (−ˆ + ˆ)

2

= +ˆ

1

= √ (+ˆ + ˆ)

ı

2

= −ˆ

= +ˆ

ı

1

ı

= √ (+ˆ − ˆ)

2

014 10.0 points

A solenoid 2.4 m long has a radius of 0.85 cm

and 900 turns. It carries a current of 4 A.

The permeability of free space is 4 π ×

10−7 T · m/A.

What is the approximate magnetic ﬁeld on

the axis of the solenoid?

Answer in units of mT.

015 (part 1 of 5) 10.0 points

A long, straight wire is in the plane of the

page and carries a current of 10 A. Point

P is also in the plane of the page and is a

perpendicular distance 0.2 m from the wire,

as shown below.

beldorin (sb37529) – Homework 8 Sources of Magnetism – catala – (20491)

5

1. in the negative x-axis

P

x

0. 2 m

10 A

2. in the negative y -axis

z

3. in the positive y -axis

y

4. in the negative z -axis

y is upward

from the paper

With reference to the coordinate system in

the ﬁgure above, what is the direction of the

magnetic ﬁeld at point P due to the current

in the wire? The permeability of free space is

4 π × 10−7 T · m/A. Gravitational eﬀects are

negligible.

1. in the positive y direction

5. in the positive z -axis

6. in the positive x-axis

017 (part 3 of 5) 10.0 points

Determine the magnitude of the magnetic

force acting on the positive particle at point

P.

Answer in units of N.

018 (part 4 of 5) 10.0 points

An electric ﬁeld is applied that causes the net

force on the positive particle to be zero at

point P .

With reference to the coordinate system

established above, what is the direction of the

electric ﬁeld at point P that could accomplish

this?

2. in the negative x direction

3. in the positive z direction

4. in the negative z direction

5. in the positive x direction

6. in the negative y direction

1. in the positive z -axis

016 (part 2 of 5) 10.0 points

A 6.64 × 10−27 kg mass particle with positive

3.2 × 10−19 C charge is initially moving parallel to the wire with a 90 m/s speed when it

is at point P , as shown below.

6.64 × 10−27 kg

90 m/s

3.2 × 10−19 C

0. 2 m

10 A

2. in the negative z -axis

3. in the negative y -axis

4. in the positive y -axis

5. in the positive x-axis

x

z

y

y is upward

from the paper

With reference to the coordinate system

established above, what is the direction of the

magnetic force acting on the positive particle

at point P ?

6. in the negative x-axis

019 (part 5 of 5) 10.0 points

Determine the magnitude of the electric ﬁeld.

Answer in units of N/C.

020 (part 1 of 2) 10.0 points

The magnetic coils of a tokamak fusion reactor are in the shape of a toroid having an inner

radius of 1.036 m and outer radius of 1.877 m.

The toroid has 1310 turns turns of wire, each

of which carries a current of 8.42 kA.

beldorin (sb37529) – Homework 8 Sources of Magnetism – catala – (20491)

Find the magnetic ﬁeld strength along the

inner radius.

Answer in units of T.

021 (part 2 of 2) 10.0 points

What is the the magnetic ﬁeld strength along

the outer radius?

Answer in units of T.

022 (part 1 of 2) 10.0 points

A magnetic ﬁeld of 1.6 T is perpendicular to

a square coil of 16 turns. The length of each

side of the coil is 7 cm.

Find the magnetic ﬂux through an individual turn of the coil.

Answer in units of mWb.

023 (part 2 of 2) 10.0 points

Find the magnetic ﬂux through an individual

turn of the coil if the magnetic ﬁeld makes an

angle of 30 ◦ with the normal to the plane of

the coil.

Answer in units of mWb.

6