Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN: 9780190698614
Author: Sadiku, Matthew N. O.
Publisher: Oxford University Press
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Chapter 9, Problem 23P

(a)

To determine

Write Maxwell’s Equations for a linear, homogeneous medium in terms of Es and Hs.

(a)

Expert Solution
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Answer to Problem 23P

The Maxwell’s Equations for a linear, homogeneous medium are ×Es=jωμHs_ and ×Hs=(σjωε)Es_.

Explanation of Solution

Calculation:

Write Maxwell’s Equations for a linear, homogeneous medium in terms of Es and Hs.

×Es=jωμHs        (1)

Here,

Es is the phasor form of electric field,

ω is the wave frequency,

μ is the permeability of the medium, and

Hs is the phasor form of magnetic field.

×Hs=(σjωε)Es        (2)

Here,

σ is the conductivity and

ε is the permittivity of the medium.

Equations (1) and (2) represent the Maxwell’s Equations for a linear, homogeneous medium.

Conclusion:

Thus, the Maxwell’s Equations for a linear, homogeneous medium are ×Es=jωμHs_ and ×Hs=(σjωε)Es_.

(b)

To determine

Write the point form of Maxwell’s Equations as eight scalar equations.

(b)

Expert Solution
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Explanation of Solution

Calculation:

Write the generalized forms of Maxwell’s four Equations.

D=ρv        (3)

B=0        (4)

×E=Bt        (5)

×H=J+Dt        (6)

Rewrite Equation (3) in point form.

(xax+yay+zaz)(Dxax+Dyay+Dzaz)=ρv

Dxx+Dyy+Dzz=ρv        (7)

Rewrite Equation (4) in point form.

(xax+yay+zaz)(Bxax+Byay+Bzaz)=0

Bxx+Byy+Bzz=0        (8)

Rewrite Equation (5) in point form.

(xax+yay+zaz)×(Exax+Eyay+Ezaz)=(Bxtax+Bytay+Bztaz)        (9)

Find (xax+yay+zaz)×(Exax+Eyay+Ezaz).

(xax+yay+zaz)×(Exax+Eyay+Ezaz)=|axayazxyzExEyEz|=[(EzyEyz)ax(EzxExz)ay+(EyxExy)az]=[(EzyEyz)ax+(ExzEzx)ay+(EyxExy)az]

Substitute [(EzyEyz)ax+(ExzEzx)ay+(EyxExy)az] for (xax+yay+zaz)×(Exax+Eyay+Ezaz) in Equation (9).

(EzyEyz)ax(EzxExz)ay+(EyxExy)az=(Bxtax+Bytay+Bztaz)

Write the scalar equations from the obtained expression.

EzyEyz=Bxt        (10)

ExzEzx=Byt        (11)

EyxExy=Bzt        (12)

Rewrite Equation (6) in point form.

(xax+yay+zaz)×(Hxax+Hyay+Hzaz)=[(Jx+Dxt)ax+(Jy+Dyt)+(Jz+Dzt)]        (13)

Find (xax+yay+zaz)×(Hxax+Hyay+Hzaz).

(xax+yay+zaz)×(Hxax+Hyay+Hzaz)=|axayazxyzHxHyHz|=[(HzyHyz)ax(HzxHxz)ay+(HyxHxy)az]=[(HzyHyz)ax+(HxzHzx)ay+(HyxHxy)az]

Substitute [(HzyHyz)ax+(HxzHzx)ay+(HyxHxy)az] for (xax+yay+zaz)×(Hxax+Hyay+Hzaz) in Equation (13).

(HzyHyz)ax+(HxzHzx)ay+(HyxHxy)az=[(Jx+Dxt)ax+(Jy+Dyt)+(Jz+Dzt)]

Write the scalar equations from the obtained expression.

HzyHyz=Jx+Dxt        (14)

HxzHzx=Jy+Dyt        (15)

HyxHxy=Jz+Dzt        (16)

Equations (7), (8), (10), (11), (12), (14), (15), and (16) represent the point of Maxwell’s Equations as eight scalar equations.

Conclusion:

Thus, the point form of Maxwell’s Equations is written.

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