Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN: 9780190698614
Author: Sadiku, Matthew N. O.
Publisher: Oxford University Press
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Chapter 7, Problem 58P
To determine

Show that the function 1R=1R=RR3.

Expert Solution & Answer
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Explanation of Solution

Calculation:

The variable R is the distance vector from the line element at the source point (x,y,z) to the field point (x,y,z) and it is expressed as,

R=|R|R=|rr|=[(xx)2+(yy)2+(zz)2]12

To find 1R:

1R=(xax+yay+zaz)(1[(xx)2+(yy)2+(zz)2]12)=(xax+yay+zaz)[(xx)2+(yy)2+(zz)2]12=(x[(xx)2+(yy)2+(zz)2]12ax+y[(xx)2+(yy)2+(zz)2]12ay+z[(xx)2+(yy)2+(zz)2]12az)=(12[(xx)2+(yy)2+(zz)2]121(2)(xx)(10)ax12[(xx)2+(yy)2+(zz)2]121(2)(yy)(10)ay12[(xx)2+(yy)2+(zz)2]121(2)(zz)(10)az)

Simplify the above equation.

1R=([(xx)2+(yy)2+(zz)2]32(xx)ax[(xx)2+(yy)2+(zz)2]32(yy)ay[(xx)2+(yy)2+(zz)2]32(zz)az)=((xx)[(xx)2+(yy)2+(zz)2]32ax+(yy)[(xx)2+(yy)2+(zz)2]32ay+(zz)[(xx)2+(yy)2+(zz)2]32az)=1[(xx)2+(yy)2+(zz)2]32[(xx)ax+(yy)ay+(zz)az]=(xx)ax+(yy)ay+(zz)az{[(xx)2+(yy)2+(zz)2]12}3

Simplify the above equation.

1R=RR3 {R=[(xx)2+(yy)2+(zz)2]12}

To find 1R:

1R=(xax+yay+zaz)(1[(xx)2+(yy)2+(zz)2]12)=(xax+yay+zaz)[(xx)2+(yy)2+(zz)2]12=(x[(xx)2+(yy)2+(zz)2]12ax+y[(xx)2+(yy)2+(zz)2]12ay+z[(xx)2+(yy)2+(zz)2]12az)=(12[(xx)2+(yy)2+(zz)2]121(2)(xx)(01)ax12[(xx)2+(yy)2+(zz)2]121(2)(yy)(01)ay12[(xx)2+(yy)2+(zz)2]121(2)(zz)(01)az)

Simplify the above equation.

1R=([(xx)2+(yy)2+(zz)2]32(xx)ax+[(xx)2+(yy)2+(zz)2]32(yy)ay+[(xx)2+(yy)2+(zz)2]32(zz)az)=((xx)[(xx)2+(yy)2+(zz)2]32ax+(yy)[(xx)2+(yy)2+(zz)2]32ay+(zz)[(xx)2+(yy)2+(zz)2]32az)=1[(xx)2+(yy)2+(zz)2]32[(xx)ax+(yy)ay+(zz)az]=(xx)ax+(yy)ay+(zz)az{[(xx)2+(yy)2+(zz)2]12}3

Simplify the above equation.

1R=RR3 {R=[(xx)2+(yy)2+(zz)2]12}

Conclusion:

Thus, the function 1R=1R=RR3 and it is shown.

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