Tutorial 4 - Webs

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18 Οκτ 2013 (πριν από 3 χρόνια και 9 μήνες)

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1

Tutorial 4



0

= 1.26 x 10
-
6

T.m/A)
, (

0

= 8.85 x10
-
12

F/m)



1.

A horizontal surface
of
area 2.5

m
2

immersed in a uniform
magnetic

field
of

magnitude
B = 42 x 10
-
6

T.

The field line
s
make an
angle of
57
o

with the
normal to the surface
, A
, as shown.

What is

the
magnetic
flux through the surface
?


Ф
B

= B X A X COS
Ө

= 42 X 10
-
6

X 2.5 X COS 57 = 57.1
μ
T
.m
2





2.

A UHF television loop antenna has a diameter of 11 cm. The magnetic field of a TV signal is normal to the plane of the loop a
nd, at
one instant of time, its magnitude is changing at a rate 0.16

T/s
. What is the
induced emf in the antenna
?


EMF = N X (DФ)/DT = N X A X dB/dt = 1 x .11
2

x π/4 x 0.16 = 1.51 x 10
-
3

v = 1.51mV








3.

The magnetic flux through the loop shown, increases according to the relation

B

= 6.0t
2

+ 7.0t
,
where

B

in mT.m
2

and

t

in
seconds. (a) What is the magnitude of

the emf induced in the loop when
t

= 2.0 s? (b) What is the direction of the current
induced
in the loop
?



| EMF | =
N X (DФ)/DT = 1 X D(6T
2
+7T) = 12T + 7


| EMF |
T=2sec

= 12 (2) + 7 = 31 mV








4.

A uniform magneti
c field is normal to the plane o
f circular loop 10 cm in diameter and made of copper wire of internal resistance
1.1 m

. At what rate must the magnetic field change with time if an induced current of 10 A is t
o appear in the loop?




D= 0.1 m ; A = 7.85 x 10
-
3

m
2

; R = 1.1 x 10
-
3

Ω

;



i
EMF = A
(dB)/dt


I x R = A dB/dt


dB/dt = ( I x R ) / A = ( 10 x 1.1 x 10
-
3

) /
7.85 x 10
-
3


dB/dt = 1.40 T/s


5.

A Gaussian surface in the shape of a right
-
circular cylinder with end caps has a radius of 12 cm and a length of 80 cm. Through
one end

th
ere is an inward magnetic flux of 25
µ
T.m
2
. At the other end there is a uniform magnetic field of 1.6 mT, normal to the
surface and directed outward. What is the net magnetic flux through the curved surface?













2

6.

The induce
d magnetic field 6.0 mm from the central axis of a circular parallel plate capacitor and between the plates is 2.0x 10
-
7
T.
The plates have radius 3.0 mm. At what rates dE/dt is the electric field between the plates changing?




=>

=>



7.

P
arallel plate capacitor has circular plates with radius R = 30 mm and plates separation distance of 5.0 mm. A sinusoidal pote
ntial
difference with a maximum value
of 150 V and a frequency of 60 Hz is applied

across the plates.

(V = 150 sin[2

(60)t]). Find the
maximum value of the induced magnetic field B
max
(R) which occurs at r = R.






8.

At what rate must the potential difference between
the plates of parallel plate capacitor with a 2.0
µF capacitance be changed to
produce a displacement current of 1.5 A?











9.

A d
iamagnetic material placed in an external

uniform magnetic field
B
ext

they develop a magnetic
dipole
moment

µ


a) opposite to the B
ext,



b) at

the same direction to B
ext,

c) perpendicular to B
ext
,


d)
µ is in

arbitrary direction.


10.

A d
iamagnetic material placed in an external non
-
uniform magnet
ic field B
ext

they

a)

attract
s

to a region of greater B
ext
,


b)
repel
s
from a region of greater B
ext,
toward a region
of lesser field



c)
repels from a region of lesser field
,


d)
develops no magnetic dipole moment
µ.



11.

A p
aramagnetic material placed in a
n external uniform magnetic field B
ext

they develops a magnetic dipole moment
µ


a) opposite to the B
ext,


b) at the same direction to B
ext,


c
) perpendicular to B
ext
,


d)
µ is in

arbitrary direction.


12.

A paramagnetic

material placed in an external non
-
un
iform magnetic field B
ext

they

a)
attracts to a region of greater B
ext
,



b)
repels from a region of greater B
ext

toward a region of lesser field


c)
repels from a region of lesser field ,


d)
develops no magnetic dipole moment
µ.



13.

A f
erromagnetic mate
rial placed in an external uniform magnetic field B
ext

they develops a strong magnetic dipole moment
µ


a) opposite to the B
ext,


b) in the direction of B
ext,


c) perpendicular to B
ext
,


d)
µ is in

arbitrary direction.



AC


3

q2

q
4

14.

A ferromagnetic
material placed i
n an external non
-
uniform magnetic field B
ext

they

a)
attracts toward a region of greater B
ext
,



b)
repels from a region of greater B
ext
toward a region of lesser field


c)
repels from a region of lesser field ,


d)
develops no magnetic dipole moment
µ
.


15.

Show that the displacement current
I
d

=

0
(d

E

/dt)

has a unit of Ampere
.


; remember that :









16.

What are the

main differences between Gauss’

law for electricity and Gauss’ law for magnetism, write down their mathematical
expressions?



17.

“A changing magnetic flux induces electric field” this statement is known by

a.

Gauss’ law for electricity,


b.

Ampere’s law for magn
etic field


c.

Faraday’s law of induction,


d.

Maxwell’s law for induced magnetic field.


e.

“A changing electric flux induces magnetic field” this statement is known by

a)

Gauss’ law for electricity,


b)

Ampere’s law for magnetic field


c)

Faraday’s law of induction,


d)

Maxwell’s law of induction.


f.

Ampere
-
Maxwell law relates

e)

induced magnetic field to changing electric flux and current,


f)

changing electric flux to induces magnetic field,

g)

changing magnetic flux to induces electric field,


h)

magnetic dipole moment
µ

to magne
tic materials.


g.

Magnetic hysteresis loop is an energy loss in ferromagnetic materials due to the effect of

i)

material’s dc high resistance
,


b)
material’s ac low impedance



j)

domain’s and domain’s wall of the material’s magnetic moments
,


k)

heat transfer

by conduction
.


21.

S
quare surface of 4.0 mm on each sides immersed in a uniform electric field with magnitude E = 2000 N/C. The field lines make
an angle of 30
o

with the normal to the surface, as shown. What is the electric flux through the surface? (

0

= 8.
85 x10
-
12

F/m)
.
CH.23


Ф
=E*A*COS(
Ө
)=2000*0.004
2
*COS(30)= 2.77 x 10
-
2
N.m
2
/C

a) 7.77 x 10
-
2
N.m
2
/C,


b) 2.77 x 10
-
2
N.m
2
/C


c) 4.77 x 10
-
2
N.m
2
/C,


d) 0.77 x 10
-
2
N.m
2
/C


22.

Four charges q
1
= 2
µ
C
,

q
2

= 3
µ
C
,

q
3
=
-
2
µ
C, q
4
=
-
3
µ
C, three neutral objects and closed Gaussian surface sho
wn in the figure. What
is the net electric flux? (

0

= 8.85 x10
-
12

F/m)
CH.23

Ф
=q
enc
/

0

= (q2+q4) /

0
= (2
µ+(
-
2
µ))

/

0

= zero

a)
-

6.8
x 10
-
2
N.m
2
/C

,

b)
-

0.565
x 10
-
2
N.m
2
/C



c) 0.68
x 10
-
2
N.m
2
/C
,

d) Zero.














23.

Infinite thin straight wire i
s carrying current of 2 A. What is the magnetic field at point outside the wire at a distance 0.5 m?

(
µ
0

= 1.26 x 10
-
6

T.m/A)



4

a) 1.8


x 10
-
6
T,

b) 0.4 x 10
-
6

T,


c) 4


x 10
-
6

T,

d) 0.8 x 10
-
6
T.


24.

Multi co
re

c
able of

three
thin wires carries the same current of 0.3 A at the
same
direction shown in the figure. What is the net
magnetic field at a distance 0.4 m out side the Amperian loop?



a) 2.4 x 10
-
6

T

b) 1.4 x 10
-
6

T,

c) 0.45 x 10
-
6

T,

d) 0.15 x
10
-
6

T


25.

Spring
-

mass linear oscillator of frequency 9.8
Hz, displacement amplitude
0.08 m and phase constant

. What is the equation of
the displacement function
X(t)

?
CH.15

X(t)=Xm*cos(wt+
Ф
)=0.08cos(2
π
9.8(t)+
π
)=0.08cos(19.6
π
t+
π
)

a) 0.16 cos (
9.8

t
-


),

b) 0.08 cos (1
9.6

t +

),


c) 0.08 cos (
9.8

t +

),

d) 0.16 cos (
9.8

t +

/2)


26.

Sound waves propagates along x axis in a fluid medium, makes the fluid’s particles oscillates
CH.17


a) perpendicular to x axis,


b) parallel to x axis


c) along z
axis,



d) making angle of 30
o

with x axis.



27.

Linear oscillator of spring constant 65 N/m and the mass position X
m

= 0.1 m. What is the mechanical energy at any time?
CH.15

E=0.5KXm
2
= 0.5 * 65 * (0.1)
2
= 0.325J

a) 3.75 J,

b) 0.325 J,


c) 3.5 J,

d)

3.05 J




28.
The displacement and velocity amplitudes are at their maximum values at the state of resonance in forced oscillations. What t
ype of
interference between the system’s angular frequency


and the ext
ernal driving force angular frequency

d
?
CH.16

l)

Fully destructive
interference
,

m)

Medium
interference
,


n)

Fully constructive
interference
,

o)

None of the answers above is correct.







29.

The angular wave number k, is defined by: number of cycles per
(CH.1
6)


a) second

b) meter,

c) kilogram,

d) watt


30.

What is the velocity of a wave,
v
, travelling along
-

x axis of a string and described by y(x,t) = 0.0037 sin(75x + 3.7t)?
CH.16

V = w/k 3.7 / 75 = 0.04933 m/s

a) 0.05 m/s


b) 0.008 m/s


c) 0.084 m/s,


d) 3.7 m/s





31.

A transverse wave y(x,t) = 0.002 sin(4x


2t) travelling along +x axis. What is the maximum transverse speed the medium
particles could have along y axis?
CH.16

U
max

| w * y
m
| = | 2 * 0.002 | = 0.004m = 4mm

a) 2 mm/s,

b) 4 mm/s,

c) 8 m
m/s,

d) 0.5 mm/s






32.

A sinusoidal wave travels a long a string. The time for a particular point to move from maximum displacement to zero is 0.17
second. What is the period (T)?
CH.16

1/4t = 0.17


t = 4 * 0.17 = 0.68s

a) 0.26 sec,

b) 0.68 sec,

c) 0
.18 sec,

d) 1.28 sec.


33.

The equation of a transverse wave travelling a long a very long string is

y = 6.0 sin(0.020

x
-

4.0

t),
where
x

and
y

are in centimetres. What is the transverse velocity at
x = 3.5 cm

when
t = 0.26
sec
.
CH.16

U = wy
m
cos(
0.020

x + 4.
0

t)= 4
π
*6cos(0.02*
π
*3.5+4*0.26*
π
) =


a)
-
23.96


cm/s,


b) 22.6


cm/s,

c)
-
1.11


cm/s,

d) 1.11


cm/s


34.

An electric dipole consists of charges
-
2q
e

and +2q
e

separated by 0.78 nm. What is their electric dipole moment?
CH.22

P = q * d = 1.6e
-
19 * 0.78e
-
9 =

2.5 x10
-
28

c.m


a) 5.5 x10
-
28

c.m,

b) 2.5 x10
-
28

c.m ,

c) 0.15 x10
-
28

c.m,

d) 0.95 x10
-
28

c.m





5

35.

A water vapour has an electric dipole moment of magnitude 6.2 x 10
-
30

C.m. What is the electric field due to this dipole at a
distance 1.1 nm from the cen
tre of the dipole
? CH.22

E = p/2
π

0

* Z
3


E = 6.2e
-
30 / ( 2 * 3.14 * 8.85e
-
12 * (1.1e
-
9)
2
=8.4 x 10
7

N/C

a) 2.4 x 10
7

N/C,


b) 5.4 x 10
7

N/C,

c) 8.4 x 10
7

N/C,


d) 12.4 x 10
7

N/C










36.

An electric dipole consists of charges of magnitude 1.5 nC separated by 6.2

m is in an electric
filed of strength 1100 N/C. What is
the maximum rotating torque acting on the dipole?
CH.
22

=1.5e
-
9*6.2e
-
6*1100=1.02 x10
-
11

N.m

a) 6.2 x10
-
12

N.m.

b) 8.2 x10
-
12

N.m

c) 10.2 x10
-
12

N.m,

d) 12.2 x10
-
12

N.m





37.

What is the fundamental fr
equency (1
st

Harmonics) for standing wave on a wire that is 10.0 m long, has a mass of 100 g, and is
stretched under a tension of 250 N?
CH.16

a) 1.9 Hz,


b) 4.9 Hz,

c) 5.9 Hz,


d) 7.9 Hz

v=
=158.11m/s

f
1
=n*v/2L = 1 x 158.11 / (2 *
10) = 7.90Hz



38.

The speed of sound wave in copper metal of density


= 8.92 x10
3

kg/m
3

is 3.56x
10
3

m/s
. If the sound wave equation [S(x,t) = (4
cm) cos(kx


2
t
)]

.

What is the sound wave intensity?
CH.17

=10.2 x 10
4

W/m
2

a) 10.2 x

10
4

W/m
2
,

b) 20.2 x 10
4

W/m
2
,


c) 30.2 x 10
4

W/m
2
,

d) 50.2 x 10
4

W/m
2




39.

Two identical sinusoidal waves, moving in the same direction along a stretched string,

y
1
(x,t) = 3.8 sin(3x
-

3t) and y
2
(x,t) = 3.8 sin( 3x


3t +

/2).

What is
the displacement equation y’(x,t) of the resultant wave due to the interference?
CH.16

Y’(x,t)=2*3.8*cos(
π
/4)sin( 3x


3t +

/4) or

Y’(x,t)=5.37(
π
/4)sin( 3x


3t +

/4).


40.

Calculate the phase difference in radian between the
shown
two waves that makes the
interference fully destructive?
CH.16

At Fully Destructive Y’
m
=0

So: cos(Ф/2) = 0

Cos
-
1
(0) = Ф/2 => Ф/2 = π/2 => Ф = π





41.

A nylon string has a linear density of 7.2 g/m and is under a tension of 150 N. The fixed supports are distance 90.0 cm a par
t and
the string is oscillating in the standing wave pattern shown in the figure.
CH.16

Calculate the

I.

Speed of the wave


II.
Wavelength of the wave.








D = 90 cm


6

q2

q4

42.

A spring mass system of mass m = 300 g, spring constant

of 9
0 N/m and damping constant

b = 140 g/s experiences damped
oscillation. How long does it take for the energy of the damped oscillations to drop to
half its initial value
?
CH.15



E=1/2 K*Xm
2
e
-
bt/m

At 1/2E:

1/2E=1/2K* Xm
2
e
-
bt/m

½(1/2K*Xm
2
)= 1/2 K*Xm
2
e
-
bt/m

(Cancel both sides)

½ = e
-
bt/m



ln(1/2)=
-
bt/m


t = (
-
ln(1/2)* m )/b

T =
(0.693 * 0.3kg)/
.140kg/s = 1.485 sec.



43.

S
quare surface of 8.0 cm on each sides immersed in a uniform electric field with magnitude E = 1000 N/C. The field lines make
an
angle o
f 30
o

with the normal to the surface, as shown. What is the electric flux through the surface?
CH.23


Ф

= E * A * cos(
ө
) = 1000 * (0.08m)
2

* cos(30
0
) = 5.54

N.m
2
/C,



a) 5.54

N.m
2
/C,


b) 2.77 N.m
2
/C


c) 4.77 N.m
2
/C,


d) 0.77 N.m
2
/C


44.

Four charges q
1
= 2
µ
C
,

q
2

= 3
µ
C
,

q
3
=
-
2
µ
C, q
4
=
-
3
µ
C, three neutral objects and closed Gaussian surface shown in the f
igure. What is
the net electric flux? (

0

= 8.85 x10
-
12

F/m)

Ф
=q
enc
/

0

= (q2+q4) /

0
= (2
µ+(
-
2
µ))

/


= zero

a)
-

6.8
x 10
-
2
N.m
2
/C

,

b)
-

0.565
x 10
-
2
N.m
2
/C



c) 0.68
x 10
-
2
N.m
2
/C
,

d) Zero.














45.

Infinite thin straight wire is carrying cu
rrent of 1.3 A. What is the magnetic field at point outside the wire at a distance 0.5 m?

(
µ
0

= 1.26 x 10
-
6

T.m/A)


a) 1.8


x 10
-
6
T,

b) 0.52 x 10
-
6

T,


c) 4


x 10
-
6

T,

d) 0.8 x 10
-
6
T.


46.

Multi core

c
ab
le of

three

thin wires carries the same current of 0.15 A at the same direction shown in the figure. What is the net magnetic
field at a distance 0.5 m out side the Amperian loop?



a) 0.18 X 10
-
6

T

b) 1.4 x 10
-
6

T,

c) 0.45 x 10
-
6

T,

d) 0.15 x 10
-
6


T





47.

If the magnetic field of a light wave oscillates parallel to a y
-
axis is given by
B
y

= B
m

sin(k
z
-


t)
, a) in what direction does the wave
travel and b) parallel to which direction does the associated electric field oscillates?



a)

-
Z => positive Z
-
Axis

b)

Positive X
-
Axis ; E
x

= E
m
sin(Kz
-
wt)


















7

E

B


48.

At a certain instant the electric and the magnetic fields directions of an electromagnetic wave
are

shown in the figure

below
. Is the wave
travelling into the page or out of it?






Using

right h
and role and known that the direction of Propagation is given by

=> the direction is
directed
into the page









49.

What is the wavelength of the electromagnetic wave emitted by the oscillator
-
antenn
a system

shown in
the
figure if L = 0.253
µ
H and C
= 25 pF?









50.

A plane electromagnetic wave has a maximum electric field of 3.2 x 10
-
4

V/m. Find the maximum magnetic field.


C = E
m

/ B
m



B
m

= E
m
/
C


3.2 x 10
-
4

/ 3 x 10
8

= 1.067 pT = 1.067 x 10
-
12

T.



51.

The electric field of a certain plane electromagnetic wave is given by
E
x

= 0, E
y
= 0, E
z

= 2.0 cos[10
15



(t


x/c)]
, with c = 3x 10
8

m/s and
all quantities in SI units. Write expressions for the co
mponents of the magnetic field of the wave.




52.

Calculate the speed of electromagnetic wave in vacuum.




53.

What inductance must be connected to a 17 pF capacitors in an oscillator capable of generating 55
0 nm electromagnetic waves?




54.

In a plane radio wave the maximum value of the
electric field component is 5.00 V/m. Calculate a) the maximum value of the magnetic
field component and
b) the
wave’s

intensity.


E
m

= 5.00 V/m , B
m

=

? , I
avg

= ?



55.

What is the intensity of travelling wave if B
m

= 1.0 x 10
-
4

T?