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 46P
To determine

To verify the Stokes’s theorem for the given region.

Expert Solution & Answer
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Explanation of Solution

Given:

The vector field F is 2ρzaρ+3zsinϕaϕ4ρcosϕaz.

The open surface is defined by the parameters, z=1,0<ρ<2,0<ϕ<45°.

Calculation:

According to the Stokes’s theorem,

  LFdl=S(×F)dS

Calculate LFdl along the contour as shown in the below figure.

Elements Of Electromagnetics, Chapter 3, Problem 46P

  LFdl=PFdl+QFdl+RFdl        (I)

For the segment P, z=1 and dl=dρaρ.

Therefore,

  F=2ρzaρ+3zsinϕaϕ4ρcosϕazF=2ρ(1)aρ+3(1)sinϕaϕ4ρcosϕazF=2ρaρ+3sinϕaϕ4ρcosϕaz

Calculate the integral (PFdl) in the segment P using the relation.

  PFdl=02(2ρaρ+3sinϕaϕ4ρcosϕaz)(dρaρ)PFdl=022ρdρPFdl=2[ρ22]02PFdl=4

For segment Q, ρ=2,z=1anddl=ρdϕaϕ.

Therefore,

  F=2ρzaρ+3zsinϕaϕ4ρcosϕazF=2(2)(1)aρ+3(1)sinϕaϕ4(2)cosϕazF=4aρ+3sinϕaϕ8cosϕaz

Calculate the integral (QFdl) in the segment Q using the relation.

  QFdl=045°(4aρ+3sinϕaϕ8cosϕaz)(ρdϕaϕ)QFdl=045°3ρsinϕdϕQFdl=3ρ[cosϕ]045°QFdl=3×2[12+1]

  QFdl=1.7573

For segment R, ϕ=45°,z=1anddl=dρaρ.

Therefore,

  F=2ρzaρ+3zsinϕaϕ4ρcosϕazF=2ρ(1)aρ+3(1)sin(45°)aϕ4ρcos(45°)azF=2ρaρ+2.12aϕ2.82az

Calculate the integral (RFdl) in the segment R using the relation.

  QFdl=20(2ρaρ+2.12aϕ2.82az)(dρaρ)QFdl=202ρdρQFdl=2[ρ22]20QFdl=4

Calculate the value of integral (LFdl) using the relation.

  LFdl=PFdl+QFdl+RFdlLFdl=4+1.75734LFdl=1.7573        (I)

Calculate the value of the curl (×F) using the relation.

  ×F=[1ρ(Fz)ϕFϕz]aρ+[FρzFzρ]aϕ+1ρ[(ρFϕ)ρFρϕ]az×F=[[1ρ(4ρcosϕ)ϕ(3zsinϕ)z]aρ+[(2ρz)z(4ρcosϕ)ρ]aϕ+1ρ[(ρ×3zsinϕ)ρ(2ρz)ϕ]az]×F=[sinϕ]aρ+[2ρ+4cosϕ]aϕ+[3zsinϕρ]az

Now calculate the integral (S(×F)dS) using the relation.

  S(×F)dS=S([sinϕ]aρ+[2ρ+4cosϕ]aϕ+[3zsinϕρ]az)(ρdϕdρaz)S(×F)dS=S3zsinϕdϕdρS(×F)dS=ρ=02ϕ=045°(3zsinϕdϕdρ)z=1S(×F)dS=3[cosϕ]045°[ρ]02

  S(×F)dS=1.758        (II)

From Equations (I) and Equation (II).

  LFdl=S(×F)dS

Thus, the Stokes’s theorem is verified.

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Chapter 3 Solutions

Elements Of Electromagnetics

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