Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN: 9780190698614
Author: Sadiku, Matthew N. O.
Publisher: Oxford University Press
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Chapter 2, Problem 7P

a)

To determine

Convert the F vector to cylindrical and spherical system.

a)

Expert Solution
Check Mark

Explanation of Solution

Given:

F=xaxx2+y2+z2+yayx2+y2+z2+4azx2+y2+z2 . (I).

Calculation:

Write the expression for the vector in Cartesian coordinates system.

  F=Axax+Ayay+Azaz        (II)

Compare the equation (I) and (II).

  Fx=xx2+y2+z2Fy=yx2+y2+z2Fz=zx2+y2+z2

Write the expression for Aρ.Aϕ, andAz in cylindrical form.

  [FρFϕFz]=[cosϕsinϕ0sinϕcosϕ0001][AxAyAz]

Substitute the obtained Ax.Ay, andAz values in the above equation.

  [FρF*Fz]=[cosϕsinϕ0sinϕcosϕ0001][xx2+y2+z2yx2+y2+z24x2+y2+z2]        (III)

Write the variable change of cylindrical to rectangular from table 2.1.

  ρ=x2+y2ρ2=x2+y2

Substitute ρ2=x2+y2        (III)

  [FρF*Fz]=[cosϕsinϕ0sinϕcosϕ0001][xρ2+z2yρ2+z24ρ2+z2]

Solve the above matrix to obtain the cylindrical coordinates.

  Fρ=1ρ2+z2[ρcos2ϕ+ρsin2ϕ]=ρρ2+z2Fo=1ρ2+z2[ρcosϕsinϕ+ρcosϕsinϕ]=0Fz=4ρ2+z2

Write the given F vector in cylindrical form.

  Fρ=1ρ2+z2[ρρ2+z2aρ+0aϕ+4ρ2+z2az]F=1ρ2+z2(ρaρ+4az)

Thus, the given vector in cylindrical form is F=1ρ2+z2(ρaρ+4az).

Similarly write the matrix for the spherical form using Table 2.1.

  [FrFθFϕ]=[sinθcosϕsinθsinϕcosθcosθcosϕcosθsinϕsinϕsinϕcosϕ0][xryr4r]

Solve the above matrix to obtain the spherical coordinates.

  Fr=r2sin2θcos2θ+rrsin2θsin2θ+4rcosθ=sin2θ+4rcosθFθ=sinθcosθcos2θ+sinθcosθ4rcosθ=sinθcosθ4rsinθFϕ=sinθcosϕsinϕ+sinθsinϕcosϕ=0

Write the given F vector in spherical form using the above equations.

  F=(sin2θ+4rsinθ)ar+sinθ(cosθ4r)aθ

Thus, the given vector in spherical form is F=(sin2θ+4rsinθ)ar+sinθ(cosθ4r)aθ.

b)

To determine

Convert the G vector to cylindrical and spherical system.

b)

Expert Solution
Check Mark

Explanation of Solution

Given:

G=(x2+y2)[xaxx2+y2+z2+yayx2+y2+z2+zaxx2+y2+z2]

Calculation:

Write the given vector G.

  G=(x2+y2)[xaxx2+y2+z2+yayx2+y2+z2+zaxx2+y2+z2]G=[x(x2+y2)axx2+y2+z2+y(x2+y2)ayx2+y2+z2+z(x2+y2)axx2+y2+z2]        (IV)

Write the variable change of spherical to rectangular from table 2.1.

  ρ=x2+y2ρ2=x2+y2

Substitute ρ2=x2+y2 in equation (IV).

  G=[x(ρ2)axρ2+z2+y(ρ2)ayρ2+z2+z(ρ2)axρ2+z2]

Write the matrix for the cylindrical form using Table 2.1.

  [GρGϕGz]=[cosϕsinϕ0sinϕcosϕ0001][x(ρ2)ρ2+z2y(ρ2)ρ2+z2z(ρ2)ρ2+z2]

Solve the above matrix to obtain the cylindrical coordinates.

  Gρ=ρ2ρ2+z2[ρcos2ϕ+ρsin2ϕ]=ρ3ρ2+z2

  Gϕ=0

  Gz=zρ2ρ2+z2

Write the given F vector in cylindrical form using the above equations.

  Gz=ρ3ρ2+z2ax+0ay+zρ2ρ2+z2azGz=ρ2ρ2+z2(ρaρ+zaz)

Thus, the given vector in cylindrical form Gz=ρ2ρ2+z2(ρaρ+zaz).

Similarly write the matrix for the spherical form using Table 2.1.

  [GrGθGϕ]=[sinθcosϕsinθsinϕcosθcosθcosϕcosθsinϕsinϕsinϕcosϕ0][xrsinθrysinθrzsinθ]

Solve the above matrix to obtain the spherical coordinates.

  Gr=rsin2θcos2ϕ+rsin2θsin2ϕ+rcos2θsinθ=rsin3θ+rcos2sinθ=rsinθGθ=rsin2θcosθcos2θ+rsin2θcos(sin3ϕ)rsin3θcosθ=rsin2θcosθrsin2cosθ=0Gϕ=rsin2θsinϕcosϕsinϕ+rsin2θcosϕsinϕ=0

Write the given F vector in spherical form using the above equations.

  G=rsinθar+0+0=rsinθar

Thus, the given vector in spherical form is G=rsinθar.

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