Tutorial 3 Solution Emagnet

TUTORIAL 3 (SOLUTION) EEE 3133 ELECTROMAGNETIC FIELDS AND WAVES DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING

Question 1 Determine the differential displacement (dL), differential normal surfaces (dS) and differential volume (dV) for the following coordinate systems. Sketch the coordinate systems to ease your solution. (a) Cartesian coordinate (b) Cylindrical coordinate (c) Spherical coordinate Refer to text book : Chapter 3 - 3.2

Question 2 The surfaces ρ = 3 and 5, ø = 100° and 130°, and z = 3 and 4.5 identify the closed surface. (a) Sketch the closed surface DIY (b) Find the volume enclosed

(c) Find the total area of the enclosing surface

(d) Find the total length of the twelve edges of the surface

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EEE 3133 Sem 1 2009/2010 prepared by Mohd Taufik

(e) Find the length of the longest straight line that lies entirely within the volume

Ans. (b) 6.28 (c) 20.7 (d) 22.4 (e) 3.21

Question 3 Find the directional derivative of T = x 2 + y 2 z along the direction xˆ 2 + yˆ 3 − zˆ 2 and evaluate it at (1,−1, 2).

Gradient of T : ∇T =  xˆ ∂ + yˆ ∂ + zˆ ∂  x 2 + y 2 z = xˆ 2 x + yˆ 2 yz + zˆy 2  ∂x ∂y ∂z  

(

)

We denote L as the given direction, L = xˆ 2 + yˆ 3 − zˆ 2

Unit vector is aˆl =

I xˆ 2 + yˆ 3 − zˆ 2 xˆ 2 + yˆ 3 − zˆ 2 = = I 17 2 2 + 32 + 2 2

and dT dl

(1, −1, 2 )

= ∇T ⋅ aˆl =

4 x + 6 yz − 2 y 2 17

= (1, −1, 2 )

− 10 17

Ans. dT dl

2

(1, −1, 2 )

EEE 3133 Sem 1 2009/2010 prepared by Mohd Taufik

= ∇T ⋅ aˆl =

4 x + 6 yz − 2 y 2 17

= (1, −1, 2 )

− 10 17

Question 4 Determine the divergence of each of the following vector fields and then evaluate it at the indicated point. (a) E = xˆ3 x 2 + yˆ 2 z + zˆx 2 z at (2 ,-2 ,0 )

∇⋅E =

∂E x ∂E y ∂E z + = 6x + 0 + x 2 = x 2 + 6x + ∂y ∂z ∂x

Thus ∇ ⋅ E (2, −2, 0 ) = 16

(b) E = Rˆ (a 3 cos θ / R 2 ) − θˆ(a 3 sin θ / R 2 ) at (a / 2 ,0 ,π )

∇⋅E =

∂ 1 ∂ 1 1 ∂Eφ 2a 3 cos θ 2 ( ) + + = − sin R E E θ R θ R sin θ ∂θ R sin θ ∂φ R 2 ∂R R3

(

)

Thus, ∇ ⋅ E (a / 2, 0,π ) = −16 Ans. (a) ∇ ⋅ E (2 , −2, 0 ) = 16 (b) ∇ ⋅ E ( a / 2 , 0,π ) = −16

Question 5 Let
       and             determine the following at point (2,-3,1). (a)





 



 



 



               

    at point (2,-3,1) (b)  

  

     

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EEE 3133 Sem 1 2009/2010 prepared by Mohd Taufik

(c) ) 

)    *

+ + + + + + ,  .  * , + + + + + +

  %     ( % ( %   (  %     ( %   (   !"#$ %& &'(

(d) 



 

/ 

/

/  

/

/  /

+%    ( +%    ( +%    ( + + +       %  ( %  (     %            (  01 !"#$ %& &'(

(e)  

           

(f) )


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Answer is 0 as the curl of the gradient of any scalar field vanishes. (g)  ) 

Answer is 0 as the divergence of the curl of any vector is zero. Ans. (a)   (b) -30 (c)  (d) 98 (e)    (f) 0 (g) 0

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EEE 3133 Sem 1 2009/2010 prepared by Mohd Taufik