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Bài giảng Electromagnetic Fields – Prob - Chapter 1 - Trần Quang Việt

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Electromagnetic Fields – Prob – ch_1  
D1.2. Give three vectors:  
A = 3a1 + 2a2 + a3;B = a1 + a2 − a3  
ꢀꢁ ꢁ  
C = a1 + 2a2 + 3a3  
ꢁ
ꢁ
ꢀꢁ ꢀꢁ ꢀꢁ  
( a ) A + B − 4C  
Find the following:  
(b) Unit vector along :  
A + 2B − C  
( c ) A.C;( d ) B×C; ( e )A.( B×C )  
Ans: (a) 13; (b)( 2a1 + a2 − 2a3 )/ 3; (c ) 10;( d ) 5a1 − 4a2 + a3  
(e) 8  
Trần Quang Việt – BMCS – Khoa Điện – ĐHBK Tp.HCM  
Electromagnetic Fields – Prob – ch_1  
D1.3.  
Three vectors A, B and C given by:  
ꢀꢁ ꢁ ꢁ ꢀꢁ ꢁ ꢀꢁ ꢁ  
A = a1 + 2a2 + 2a3;B = 2a1 + a2 − 2a3;C = a1 − a2 + a3  
ꢁ
ꢁ
ꢁ
ꢁ
ꢁ
Find the following: ( a) A×( B×C ); (b) B×(C × A) and  
( c) C×( A× B )  
ans : (a) 2a1 + a2 − 2a3;(b ) a1 + 2a2 + 2a3;  
ꢁ
( c ) -3a1 − 3a2  
ꢁ
Trần Quang Việt – BMCS – Khoa Điện – ĐHBK Tp.HCM  
1
Electromagnetic Fields – Prob – ch_1  
D1.4. Three points P1, P2 and P3 are given by (1, -2,2), (3,1,0), and  
(5,2,-2) respectively. Obtain the following: (a) the vector  
drawn from P1 to P2; (b) the straight-line distance from P2 to  
P3; and (c) the unit vector along the line from P1 to P3  
ans : (a) 2ax + 3ay − 2az ;(b ) 3;(c) (ax + ay − az )/ 3  
Convert into cyclindrical coordinates the following points  
D1.8.  
specified in cartesian coordinates : (a) (-2,0,1);  
(b ) (1,- 3,-1); and (c) (- 2,- 2,3)  
ans :( a ) ( 2,π ,1); (b ) ( 2,5π / 3,−1); (c ) ( 2,5π / 4,3)  
Trần Quang Việt – BMCS – Khoa Điện – ĐHBK Tp.HCM  
Electromagnetic Fields – Prob – ch_1  
Convert into spherical specified in cartesian coordinates :  
D1.9.  
(a) (0,-2,0);  
(b ) (-3, 3,2); and (c) (- 2,0,- 2)  
ans :( a ) ( 2,π / 2,3π / 2 );  
(b ) ( 4,π / 3,5π / 6); ( c ) ( 2,3π / 4,π )  
Three unit vectors are given in cyclindrical coordinates as  
follows:  
P1.17.  
A = ar at ( 2,π / 6,0 ),B = aφ at (1,π / 3,2 )  
and C = aφ at (3,5π / 6,1). Find: (a) A.B,(b) B.C  
and (c) B×C  
Trần Quang Việt – BMCS – Khoa Điện – ĐHBK Tp.HCM  
2
Electromagnetic Fields – Prob – ch_1  
Three unit vectors are given in spherical coordinates as  
follows:  
P1.17.  
A = ar at ( 2,π / 6,π / 2 ),B = aθ at (1,π / 3,0 )  
and C = aφ at (3,π / 4,3π / 2 ). Find: (a) A.B,(b) B.C  
(c) A.C, and (d) A× B.C  
Find the outward pointing unit vectors normal to the closed  
surface 2x2+2y2+z2=8 at the following points:  
D5.1.  
( a ) ( 2, 2,0); (b) (1,1,2); and (c) (1, 2, 2)  
ans : (a) (ax + ay )/ 2, (b) (ax + ay + az )/ 3  
ꢁ
ꢁ
ꢁ
(c) ( 2ax + ay + az )/ 7  
Trần Quang Việt – BMCS – Khoa Điện – ĐHBK Tp.HCM  
Electromagnetic Fields – Prob – ch_1  
D5.2. Two scalar functions are given by:  
Φ1( x,y,z ) = x2 + y2 + z2  
Φ2( x,y,z ) = x + 2y + 2z  
Find the following at the point (3,4,12) : (a) the maximum rate of  
increase of Φ1, (b) the maximum rate of increase of Φ2; and (c) the  
maximum rate of increase of Φ1 along the direction of the  
maximum rate of increase of Φ2  
ans : (a) 26, (b) 3, (c) 2313  
Trần Quang Việt – BMCS – Khoa Điện – ĐHBK Tp.HCM  
3
Electromagnetic Fields – Prob – ch_1  
P3.1. Find the curls of the following vectors fields:  
( a ) zxax + xyay + yzaz  
ꢁ
(b ) cos( y )ax − xsin( y )ay  
ꢁ
ꢁ
2
( c ) e−r / r aφ in cyclindrical coordinates  
(
( d ) 2r cosθ ar +raθ in spherical coordinates  
)
ꢁ
ꢁ
Trần Quang Việt – BMCS – Khoa Điện – ĐHBK Tp.HCM  
Electromagnetic Fields – Prob – ch_1  
P3.8. Find the divergences of the following vectors fields:  
( a ) zxax + xyay + yzaz  
ꢁ
(b ) 3ax +( y −3)ay +( 2 + z )az  
ꢁ
ꢁ
ꢁ
( c ) r sinφaφ in cyclindrical coordinates  
ꢁ
( d ) rcosθ(cosθ ar - sinθaθ ) in spherical coordinates  
ꢁ
Trần Quang Việt – BMCS – Khoa Điện – ĐHBK Tp.HCM  
4
Electromagnetic Fields – Prob – ch_1  
D1.16.  
Infinite plane sheets of charge lie in the z=0, z=2, and z=4  
planes with uniform surface charge densities ρs1, ρs2 and ρs3  
respectively. Given that the resulting electric field intensities  
at the points (3,5,1), (1,-2,3), and (3,4,5) are  
0, 6az , and 6az  
V/m respectively, find the following: (a) ρs1, (b) ρs2, (c) ρs3  
E at (-2,1,-6)  
and (d)  
ans : (a) 4ε0 (C/m2 );(b ) 6ε0 (C/m2 );(c) -2ε0 (C/m2 );  
ꢁ
(d) -4az (V / m )  
Trần Quang Việt – BMCS – Khoa Điện – ĐHBK Tp.HCM  
Electromagnetic Fields – Prob – ch_1  
D1.20.  
Infinite plane sheets of current lie in the x=0, y=0, and z=0  
planes with uniform surface current densities :  
Js0 az , 2Js0 ax , and -Js0 ax (A/m)  
respectively. Find the  
resulting magnetic flux densities at the following points :  
(a) (1,2,2), (b) (2,-2,-1), and (c) (-2, 1,-2)  
ans : (a) µ0Js0 (ay +az );(b ) -µ0Js0 az ;(c) µ0Js0 (-ay +az )  
Trần Quang Việt – BMCS – Khoa Điện – ĐHBK Tp.HCM  
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