Elements of Electromagnetics
Elements of Electromagnetics
7th Edition
ISBN: 9780190698669
Author: Sadiku
Publisher: Oxford University Press
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Chapter 4, Problem 30P

(a)

To determine

The charge density.

(a)

Expert Solution
Check Mark

Explanation of Solution

Given:

The electric flux density D is 2ρ(z+1)cosϕaρρ(z+1)sinϕaϕ+ρ2cosϕazμC/m2.

Calculation:

Calculate the volume charge density (ρV) using the relation.

  ρV=D=1ρρ(ρDρ)+1ρϕ(Dϕ)+z(Dz)

  ρV={1ρρ[2ρ2(z+1)]+1ρϕ[ρ(z+1)sinϕ]+z[ρ2cosϕ]}

  ρV={1ρ[4ρ(z+1)]+1ρ(ρ(z+1)cosϕ)+0}μC/m3

  ρV=[4(z+1)cosϕ(z+1)cosϕ]μC/m3ρV=3(z+1)cosϕμC/m3

Thus, the charge density is 3(z+1)cosϕμC/m3.

(b)

To determine

The total charge enclosed by the volume 0<ρ<2,0<ϕ<π/2,0<z<4.

(b)

Expert Solution
Check Mark

Explanation of Solution

Given:

The electric flux density D is 2ρ(z+1)cosϕaρρ(z+1)sinϕaϕ+ρ2cosϕazμC/m2.

Calculation:

Take the elemental volume (dv) as ρdρdϕdz.

Calculate the total charge (Q) using relation.

  Q=VρVdv

  Q=V(2ρ(z+1)aρρ(z+1)sinϕaϕ+ρ2cosϕaz)ρdρdϕdz=040π/202(3(z+1)cosϕμC/m3)ρdρdϕdz=(3μC)[ρ22]02[z22+z]04[sinϕ]0π/2=72μC

Thus, the total charge enclosed by the volume 0<ρ<2,0<ϕ<π/2,0<z<4 is 72μC.

(c)

To determine

The net flux through the surface of the volume in (b).

(c)

Expert Solution
Check Mark

Explanation of Solution

Given:

The electric flux density D is 2ρ(z+1)cosϕaρρ(z+1)sinϕaϕ+ρ2cosϕazμC/m2.

Calculation:

The net flux in cylinder does not vary with z, the flux from top to bottom is equal and opposite in nature.

Calculate the net flux (Ψ) using the relation.

  Ψ=SDdS

  Ψ=S(2ρ(z+1)cosϕaρμC/m2ρ(z+1)sinϕaϕμC/m2+ρ2cosϕazμC/m2)ρdϕdzaρ=0π204(2(2)(z+1)cosϕ)(2μC/m2)dzdϕ=(8μC)[z22+z]04[sinϕ]0π2=(8μC)[4202+40][sinπ2sin0]    Ψ=(8μC)(12)(10)=96μC

Calculate the flux density through the flat surfaces as a component of ϕ=0° using the relation.

  D1=ρ(z+1)sinϕ

  D1=ρ(z+1)sin0°=0

Calculate the flux density through the flat surfaces as component of ϕ=90° using the relation.

  D2=ρ(z+1)sinϕ

  D2=ρ(z+1)sin90°=ρ(z+1)

Calculate the flux through the flat surface (Ψ) at ϕ=0° using the relation.

  Ψ2=0204ρ(z+1)dρdz

    Ψ2=[z22+z]04[ρ22]02=[1602+40][402]=24μC

Calculate the net flux (Ψnet) through the surface using the relation.

  Ψnet=Ψ1+Ψ2

  Ψnet=96μC+(24μC)=72μC

Thus, the net flux through the surface of the volume is 72μC.

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Elements of Electromagnetics

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