Correct answer: 1
.
00624
×
10

20
N.
Explanation:
Lorentz’s Force
~
F
=
q
(
~v
×
~
B
+
~
E
)
.
Faraday’s Law in general form
E
=
I
~
E
·
d~s
=
d
Φ
B
dt
.
Platt, David – Homework 22 – Due: Oct 31 2005, 4:00 am – Inst: Ken Shih
4
Lorentz’s Force
~
F
=
q
(
~v
×
~
B
+
~
E
)
.
at
P
, there is only induced electric field and
no magnetic field, so Lorentz’s equation can
be simplified as
~
F
=
q
~
E .
To calculate electric field at
P
, note that
the magnetic field region is circular, so there
is a symmetry for the induced electric field,
i.e.
, they are equal magnitudely for all points
on a circle of radius
r
away from the center of
the magnetic field; so
I
~
E d~s
= 2
π r E ,
while
d
Φ
B
dt
=
d
(
B
·
A
)
dt
=
d
(
B
·
π R
2
)
dt
=
π R
2
d B
dt
=
π R
2
(3
d t
2

2
a t
)
.
From Faraday’s Law in general form, we get
2
π r E
=
πR
2
(
3
d t
2

2
a t
)
;
i.e.
,
E
=
R
2
2
r
(
3
d t
2

2
a t
)
=
(0
.
018 m)
2
2 (0
.
0474 m)
h
3 (2 T
/
s
3
) (2
.
49 s)
2

2 (3
.
78 T
/
s
2
) (2
.
49 s)
i
= 0
.
0628047 N
/
C
,
and so the magnitude of the force exerted on
the electron is
F
=
q
e
E
= (1
.
60218
×
10

19
C) (0
.
0628047 N
/
C)
= 1
.
00624
×
10

20
N
k
~
F
k
=
1
.
00624
×
10

20
N
.
007
(part 2 of 3) 10 points
And direction of the force is
1.
same as that of the magnetic field.
2.
opposite to that of the electric field.
cor
rect
3.
opposite to that of the magnetic field.
4.
same as that of the electric field.
5.
not determinable due to insufficient infor
mation.
6.
perpendicular to that of the electric field
and magnetic field.
Explanation:
The direction of the force is opposite to that
of the electric field
~
E
, since the charge on the
electron is negative.
008
(part 3 of 3) 10 points
At what (nonzero) time is this force equal to
zero?
Correct answer: 1
.
26 s.
Explanation:
The
F
= 0 if and only if the electric field is
zero. From explanation for Part 1 we know
E
=
R
2
2
r
d B
dt
from this we get the condition for the
F
= 0
is
d B
dt
= 0
i.e.
,
3
d t
2

2
a t
= 0
So,
t
= 0 s
or
t
=
2
a
3
d
=
2 (3
.
78 T
/
s
2
)
3 (2 T
/
s
3
)
=
1
.
26 s
.
The
t
= 0 solution is trivial!
Platt, David – Homework 23 – Due: Nov 2 2005, 4:00 am – Inst: Ken Shih
1
This
printout
should
have
7
questions.
Multiplechoice questions may continue on
the next column or page – find all choices
before answering.
The due time is Central
time.
001
(part 1 of 1) 10 points
A circuit consists of a 32 Ω resistor and a
740 mH inductor connected in series to a 17 V
battery.
What is the value of the current when the
current is increasing at the rate of 4 A
/
s?
Correct answer: 0
.
43875 A.
Explanation:
Let :
R
= 32 Ω
,
L
= 740 mH = 0
.
74 H
,
V
= 17 V
,
and
Δ
i
Δ
i
= 4 A
/
s
.
In an RL circuit
V
=
L
Δ
i
Δ
t
+
i R
i
=
V

L
Δ
i
Δ
t
R
i
=
17 V

(740 mH) (4 A
/
s)
32 Ω
i
= 0
.
43875 A
002
(part 1 of 1) 10 points
In the circuit below a steady current
I
0
is
flowing when the switch is in position “a”. If
the switch is then thrown to position “b”, the
current will decrease to zero after some time.
L
R
E
S
b
a
I
(
t
)
Suppose we replace the inductor with one of
the same area and length, but twice as many
turns.