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Bài giảng Electromagnetic Fields – Prob - Chapter 3 - Trần Quang Việt

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Electromagnetic Fields – Prob – ch_3  
E = E0 cos(6π .108t − 2π z )ax (V / m )  
D3.1. Given :  
find the time rate of increase of By at t=10-8s for each of the  
following values of z : (a) 0; (b) ¼ m; and (c) 2/3 m.  
Ans :( a ) 0; (b) 2πE0 ; ( c )− 3π E0  
8
2
H = H0e−( 3.10 t−z ) ay ( A/ m )  
D3.3. Given :  
and :  
J = 0  
find the time rate of increase of Dx for each of the following cases :  
(a) z=2m, t=10-8s; (b) z=3m, t=(1/3).10-8s; and (c) z=3 m, t=10-8s.  
Ans :( a ) -0.7358H0; (b) 0.0733H0 ; ( c ) 0  
Trần Quang Việt – BMCS – Khoa Điện – ĐHBK Tp.HCM  
Electromagnetic Fields – Prob – ch_3  
D3.6. In a small region around the origin, the current density due to  
2
2
2
2
flow of charges is given by :  
J = J0( x ax +y ay +z az ) ( A/ m )  
where J0 is a constant. Find the time rate of increase of the charge  
density at each of the following points: (a) (0.02, 0.01, 0.01); (b)  
(0,02, -0.01, -0.01); and (c) (-0.02, -0.01, 0.01).  
C
m3s  
C
m3s  
Ans :( a ) -0.08J0 (  
); (b) 0; ( c )0.04J0(  
)
D3.15.The magnetic field of a uniform plane wave in free space is  
8
H = H0 cos(6π .10 t + 2π y )ax (A/m)  
given by:  
Find unit vector along the following: (a) the direction of propagation  
of the wave; (b) the direction of the magnetic field at t=0, y=0; and (c)  
the direction of the electric field at t=0, y=0  
Ans :( a ) -ay ; (b) ax ; (c ) az  
Trần Quang Việt – BMCS – Khoa Điện – ĐHBK Tp.HCM  
1
Electromagnetic Fields – Prob – ch_3  
D3.19.The magnetic field associated with a uniform plane wave  
propagating in the +z-direction in free space is given by:  
H = H0 cos(6π .107 t − 0.2π z )ay (A/m)  
Find the following : (a) the instantaneous power flow across a surface  
of area 1m2 in z=0 plane at t=0; (b) the instantaneous power flow  
across a surface of area 1m2 in z=0 plane at t=(1/8) µs; and (c) the  
time-average power flow across a surface of area 1m2 in z=0 plane.  
Ans :( a ) 120π H02 (W ); (b) 0; ( c ) 60π H02 (W )  
Trần Quang Việt – BMCS – Khoa Điện – ĐHBK Tp.HCM  
Electromagnetic Fields – Prob – ch_3  
E = E0e−αz cosωtax in free space  
P3.4. For the electric field  
(
), find B that satisfies Faraday’s law in differential form  
J = 0  
and the determine if the pair of  
and  
satisfy Ampere’s  
B
E
circuital law in differential form.  
P3.25. The electric-field intensity of a uniform plane wave  
propagating in free space is given by:  
E = 37.7cos(9π .107 t + 0.3π y )ax (V / m )  
Find : (a) the frequency; (b) the wavelength; (c) the direction of  
propagation of the wave; and (d) the asssiated magnetic field  
intensity vector  
H
Trần Quang Việt – BMCS – Khoa Điện – ĐHBK Tp.HCM  
2
Electromagnetic Fields – Prob – ch_3  
P3.37. The electric and magnetic fields in a coaxial cable, an  
arrangement of two coaxial perfectly conducting cylinders of radius a  
and b (>a), are given by:  
ꢀꢁ  
E =  
ꢁ
cosω(t − z µ0ε0 )ar for a < r < b  
V0  
rln(b / a )  
I0  
ꢀꢀꢁ  
H =  
ꢁ
cosω(t − z µ0ε0 )aφ for a < r < b  
2πr  
where V0 and I0 are constants and the axis of the cylinders is the z-  
axis. (a) Find the instantaneous and time-average Poyting vectors  
assosiated with the fields. (b) Find the time-average power flow  
along the coaxial cable.  
Trần Quang Việt – BMCS – Khoa Điện – ĐHBK Tp.HCM  
3
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