Bài giảng Electromagnetic Fields - Chapter 1 - Trần Quang Việt

Nội dung text: Bài giảng Electromagnetic Fields - Chapter 1 - Trần Quang Việt

1. Electromagnetic Fields – Prob – ch_1
      
      D1.2. Give three vectors: A=3 a1 + 2 a 23 + a;Ba =+− 123 a a
       Ca=1 +2 a 2 + 3 a 3
   
   
    Find the following: ( a ) A+ B − 4C
   
   
    (b) Unit vector along : A+ 2B − C
   
   
   
   
   
   
    (c) A.C;(d) B× C;(e)A.(B × C)   Ans : (a) 13; (b) (a2123+− a 2 a)/;(c) 3 10 ;(d)a 5 123 −+ 4 a a (e) 8  Tr ần Quang Vi ệt – BMCS – Khoa Điện – ĐHBK Tp.HCM Electromagnetic Fields – Prob – ch_1
   
   
    D1.3. Three vectors A, B and C given by:
     
      
      Aa=++123 22 a a;B =+− 2 a 123 a 2 a;Ca =−+ 123 a a
   
   
   
   
   
    Find the following: (a ) A×× (BC); (b) B ×× (C A) and
   
   
    (c) C × (A × B)     ans: (a) 2 a12+− a 2 a;(b)a 3 1 ++ 2 a 2 2 a; 3   (c) -3 a1− 3 a 2  Tr ần Quang Vi ệt – BMCS – Khoa Điện – ĐHBK Tp.HCM 1
2. Electromagnetic Fields – Prob – ch_1 P1.17. Three unit
    vectors  are given in spherical
    coordinates  as follows: A= aat(,r 2π / 6 , π / 2 ),B = aθ at(, 1 π /,) 30
    
   
   
   
    and C= aat(,/,φ 3π 4 3 π /). 2 Find: (a) A .B, (b) B .C
   
   
   
   
    (c) A.C, and (d) A× B .C D5.1. Find the outward pointing unit vectors normal to the closed surface 2x 2+2y 2+z 2=8 at the following points: ( a ) ( 2, 2,0); (b) (1,1,2);  and (c) (1, 2, 2) ans : (a) (axy+ a )/ 2 , (b) (a xyz + a + a ) / 3    (c) ( 2ax+ a y + a z )/ 7  Tr ần Quang Vi ệt – BMCS – Khoa Điện – ĐHBK Tp.HCM Electromagnetic Fields – Prob – ch_1 D5.2. Two scalar functions are given by: 2 2 2 Φ1(x,y,z)= x + y + z Φ2(x,y,z)= x +2 y + 2 z 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 1 ans : (a) 26 , (b) 3 , (c) 23 3  Tr ần Quang Vi ệt – BMCS – Khoa Điện – ĐHBK Tp.HCM 3
3. 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 6a z V/m respectively,
    find the following: (a) ρs1 , (b) ρs2 , (c) ρs3 and (d) E at (-2,1,-6) ans : (a) 4ε (C/m2 ) ;( b ) 6 ε (C/m 2 ) ; (c) -2 ε (C/m 2 ); 0 0 0 (d) -4az ( V /m)  Tr ần Quang Vi ệt – 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 , 2J s0 a x , and -J s0 a x (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) µ0 J s0 (a+a)yz ;(b ) - µ 0 J s0 a z ; (c) µ 0 J s0 (-a+a) yz  Tr ần Quang Vi ệt – BMCS – Khoa Điện – ĐHBK Tp.HCM 5