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 10P

a)

To determine

Convert the vector A from cylindrical to rectangular coordinate system.

a)

Expert Solution
Check Mark

Explanation of Solution

Given:

  A=ρsinϕap+ρcosaϕ+ρcosaz

Calculation:

Write the given vector A.

  A=ρsinϕap+ρcosϕaϕ2zaz

Here, Aρ=ρsinϕ, Aϕ=ρcosϕ, and Az=2z.

Write the expression for the transformation of vector A from (Ax,Ay,Az)to(Aρ,Aϕ,Az).

  [AxAyAz]=[cosϕsinϕ0sinϕcosϕ0001]1[AρAϕAz]=[ρsinϕcosϕρcosϕsinϕρsin2ϕ+ρcos2ϕ2z]=[0ρ(sin2ϕ+cos2ϕ)2z][AxAyAz]=[0ρ2z]        (I)

Write the relationship between cylindrical to rectangular.

  ρ=x2+y2z=z

Substitute ρ=x2+y2 and z=z in equation (I).

  [AxAyAz]=[0x2+y22z]A=x2+y2ay2zaz

Thus, the vector A in rectangular coordinate system is A=x2+y2ay2zaz_.

b)

To determine

Convert the vector B from spherical to rectangular coordinate system.

b)

Expert Solution
Check Mark

Explanation of Solution

Given:

  B=4rcosϕar+raθ        (II)

Calculation:

Write the given vector B.

  B=4rcosϕar+raθ

Here, Br=4rcosϕ,Bθ=r, andBϕ=0.

  [BxByBz]=[sinθcosϕcosθcosϕsinϕsinθsinϕcosθsinϕcosϕcosθsinθ0][4rcosϕr0][BxByBz]=[4rcosϕcosϕsinθ+rcosθcosϕ4rcosϕsinϕsinθ+rcosθsinϕ4rcosϕcosθrsinθ][BxByBz]=[4rcos2ϕsinθ+rcosθcosϕ4rcosϕsinϕsinθ+rcosθsinϕ4rcosϕcosθrsinθ]        (III)

Write the relationship between spherical to rectangular.

  r=x2+y2+z2θ=tan1x2+y2zϕ=tan1yx

Calculate the sinθ components.

  sinθ=ρr=x2+y2x2+y2+z2

Calculate the cosθ components.

  cosθ=zr=x2+y2x2+y2+z2

Calculate the sinϕ components.

  sinϕ=yρ=yx2+y2

Calculate the cosϕ components.

  cosϕ=xρ=xx2+y2

Calculate the x component of B vector from the equation (II).

  Bx=4rcos2ϕsinθ+rcosθcosϕ={4(x2+y2+z2)(xx2+y2)2(x2+y2x2+y2+z2)+(x2+y2+z2)(zx2+y2+z2)(xx2+y2)+}Bx=4x2x2+y2+zxx2+y2

Calculate the y component of B vector from the equation (II).

  By=4rcosϕsinϕsinθ+rcosθsinϕ={4(x2+y2+z2)(xx2+y2)(yx2+y2)(x2+y2x2+y2+z2)+(x2+y2+z2)(zx2+y2+z2)(yx2+y2)}By=4xyx2+y2+zyx2+y2

Calculate the z component of B vector from the equation (II).

  Bz=4rcosϕcosθrsinθ={4(x2+y2+z2)(xx2+y2)(zx2+y2+z2)(x2+y2+z2)(x2+y2x2+y2+z2)}Bz=4xzx2+y2x2+y2

Substitute the component of Vector B in the equation (II).

  B=1x2+y2((4x2+zx)ax+(4xy+zy)ay+(4xzx2y2)az).

Thus, the vector B in rectangular coordinates system is B=1x2+y2((4x2+zx)ax+(4xy+zy)ay+(4xzx2y2)az)_.

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