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 49P

(a)

To determine

The integral LQdl.

(a)

Expert Solution
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Explanation of Solution

Given:

The slant height of the cone (l) is 2m.

The cone angle (θ) is 30°.

The vector field (Q) is x2+y2+z2x2+y2[(xy)ax+(x+y)ay].

Calculation:

Calculate the radius of the cone (r) using the relation.

  r=lsin30°=(2m)sin30°=1m

Calculate the height (h) of the cone using the relation.

  h=lcos30°=(2m)×32=3m

Write the equation of the cone using the relation.

  x2+y2=(rh)2z2x2+y2=(13)2z2z2x2+y2=3

Substitute the value in the vector field (Q) using the relation.

  Q=x2+y2+z2x2+y2[(xy)ax+(x+y)ay]=1+z2[(xy)ax+(x+y)ay]=1+3[(xy)ax+(x+y)ay]=2[(xy)ax+(x+y)ay]

Calculate the integral (LQ.dl) using the relation.

  LQdl=L2[(xy)ax+(x+y)ay](dxax+dyay+dzaz)=2(xy)dx+(x+y)dy=2[x(xy)y(x+y)]ds=22ds

  =4ds=4(πr2)=4(π(1)2)=4π

Thus, the value of the integral is 4π_.

(b)

To determine

The integral s1(×Q)dS.

(b)

Expert Solution
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Explanation of Solution

Calculation:

Calculate the curl (×Q) using the relation.

  ×Q=|axayazxyz2(xy)2(x+y)0|=4az

Calculate the integral (s1(×Q)ds) using the relation.

  s1(×Q)ds=(4ax)ds=(4ax)dxdyaz=4dxdy=4(2π(1m)2)=8π

Thus, the value of the integral is 8π_.

(c)

To determine

The integral s2(×Q)dS.

(c)

Expert Solution
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Explanation of Solution

Calculate the integral using the relation.

s2(×Q)ds=(4az)(azdxdy)=4dxdy=4(πrl)=4(π×1×2)=8π

Thus, the value of the integral is 8π_.

(d)

To determine

The integral s1QdS.

(d)

Expert Solution
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Explanation of Solution

Calculate the integral (s1QdS) using the relation.

  s1QdS=[(xy)ax+(x+y)ay]dV=[x(x+y)y(xy)]dV=2dV=2×23×π×(1)3=4.186

Thus, the integral s1QdS is 4.186_.

(e)

To determine

The integral s2QdS.

(e)

Expert Solution
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Explanation of Solution

Calculate the integral (s2QdS) using the relation.

  s2QdS=[(xy)ax+(x+y)ay]dV=2[x(x+y)y(xy)]dV=2dV=2×(13×π×(1)2×3)=3.6276

Thus, the integral s2QdS is 3.6276_.

(f)

To determine

The integral vQdV.

(f)

Expert Solution
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Explanation of Solution

Calculate the integral (vQdV) using the relation.

  vQdV=v2[(xy)ax+(x+y)ay]dV=2vdV=2[23π(1)2+13π(1)23]=7.8164

Thus, the integral (vQdV) is 7.8164_.

The result in part (a) is the line integral of the base of cone, whereas the result in part (f) includes the volume integral about the combine shape i.e. hemi-sphere and the cone.

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Chapter 3 Solutions

Elements Of Electromagnetics

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