Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN: 9780190698614
Author: Sadiku, Matthew N. O.
Publisher: Oxford University Press
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Chapter 3, Problem 31P

(a)

To determine

The value of (r)T.

(a)

Expert Solution
Check Mark

Explanation of Solution

Given:

The vector (T) is 2zyax+xy2ay+x2yzaz.

The position vector (r) is xax+yay+zaz.

Calculation:

Calculate the value of (r)T using the relation.

  (r)T=((xax+yay+zaz)r)T

  (r)T=((xax+yay+zaz)(xax+yay+zaz))(2zyax+xy2ay+x2yzaz)=(x(x)+y(y)+z(z))(2zyax+xy2ay+x2yzaz)=(3)(2zyax+xy2ay+x2yzaz)=6zyax+3xy2ay+3x2yzaz

Thus, the value of (r)T is 6zyax+3xy2ay+3x2yzaz_.

(b)

To determine

The value of (r)T.

(b)

Expert Solution
Check Mark

Explanation of Solution

Given:

The vector (T) is 2zyax+xy2ay+x2yzaz.

The position vector (r) is xax+yay+zaz.

Calculation:

Calculate the value of (r)T using the relation.

  (r)T=(r(xax+yay+zaz))T

  (r)T=((xax+yay+zaz)(xax+yay+zaz))(2zyax+xy2ay+x2yzaz)=((x)x+(y)y+(z)z)(2zyax+xy2ay+x2yzaz)={(x)(2zyax+xy2ay+x2yzaz)x+(y)(2zyax+xy2ay+x2yzaz)y+(z)(2zyax+xy2ay+x2yzaz)z={(x)((0)ax+(y2)ay+(2xyz)az)+(y)((2z)ax+(2xy)ay+(x2z)az)+(z)((2y)ax+(0)ay+(x2y)az)

  (r)T=(xy2ay+2x2yzaz)+(2yzax+2xy2ay+x2yzaz)+(2yzax+x2yzaz)=4yzax+3xy2ay+4x2yzaz

Thus, the value of (r)T is 4yzax+3xy2ay+4x2yzaz_.

(c)

To determine

The value of r(rT).

(c)

Expert Solution
Check Mark

Explanation of Solution

Given:

The vector (T) is 2zyax+xy2ay+x2yzaz.

The position vector (r) is xax+yay+zaz.

Calculation:

Calculate the value of r(rT) using the relation.

  r(rT)=(xax+yay+zaz)r(rT)

  r(rT)={((xax+yay+zaz)(xax+yay+zaz))((xax+yay+zaz)(2zyax+xy2ay+x2yzaz))=(x(x)+y(y)+z(z))((x)(2zy)+(y)(xy2)+(z)(x2yz))=(3)(2xzy+xy3+x2yz2)=6xzy+3xy3+3x2yz2

Thus, the value of r(rT) is 6xzy+3xy3+3x2yz2_.

(d)

To determine

The value of (r)r2.

(d)

Expert Solution
Check Mark

Explanation of Solution

Given:

The vector (T) is 2zyax+xy2ay+x2yzaz.

The position vector (r) is xax+yay+zaz.

Calculation:

Calculate scalar of position vector (r) using the relation.

  |r|=x2+y2+z2

Calculate the magnitude of square of position vector (r2) using the relation.

  r2=|r|2

  r2=(x2+y2+z2)2=(x2+y2+z2)

Calculate the value of (r)r2 using the relation.

  (r)r2=(r(xax+yay+zaz))r2

  (r)r2=((xax+yay+zaz)(xax+yay+zaz))(x2+y2+z2)=((x)x+(y)y+(z)z)(x2+y2+z2)=(x)(x2+y2+z2)x+(y)(x2+y2+z2)y+(z)(x2+y2+z2)z=(x)(2x)+(y)(2y)+(z)(2z)

  (r)r2=2x2+2y2+2z2

Thus, the value of (r)r2 is 2x2+2y2+2z2_.

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Chapter 3 Solutions

Elements Of Electromagnetics

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