Elements of Electromagnetics
Elements of Electromagnetics
7th Edition
ISBN: 9780190698669
Author: Sadiku
Publisher: Oxford University Press
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Chapter 9, Problem 40P
To determine

Prove that the given magnetic vector potential satisfies the expression 2Aμε2At2=μJ in free space and find the corresponding electric scalar potential.

Expert Solution & Answer
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Answer to Problem 40P

The given magnetic vector potential satisfies the expression 2Aμε2At2=μJ in free space and the electric scalar potential is cosθj4πωεor(jβ+1r)ej(ωtβr)_.

Explanation of Solution

Calculation:

Write the expression has to be satisfied by the given magnetic vector potential.

2Aμε2At2=μJ

Rewrite the expression for free space region.

2Aμoεo2At2=μo(0) {J=0}2A=μoεo2At2

Substitute μo4πr[cos(θ)arsin(θ)aθ]ejω(trc) for A on the right side of the expression.

2A=μoεo2t2{μo4πr[cos(θ)arsin(θ)aθ]ejω(trc)}=μoεot(t{μo4πr[cos(θ)arsin(θ)aθ]ejω(trc)})=μoεot(jω{μo4πr[cos(θ)arsin(θ)aθ]ejω(trc)})=μoεo(jω)(jω){μo4πr[cos(θ)arsin(θ)aθ]ejω(trc)}

Simplify the expression.

2A=μoεoω2A {μo4πr[cos(θ)arsin(θ)aθ]ejω(trc)=Aj2=1}=β2A {β2=μoεoω2}

Therefore, the expression 2A=β2A has to be satisfied by the given magnetic vector potential.

2A=β2A        (1)

Rewrite the given magnetic vector potential.

A=μo4πrejωtej(ωc)r[cos(θ)arsin(θ)aθ]=μo4πrejωtejβr[cos(θ)arsin(θ)aθ] {ωc=β}=μo4πrejωtejβraz=μo4πejωt(ejβrr)az

Consider ejβrr as R and rewrite the expression.

A=μo4πejωtRaz        (2)

Find 2R.

2R=1r2sin(θ){r[r2sin(θ)Rr]}=1r2{r(r2)(jβr1r2)ejβr} {R=ejβrr}=1r2(β2r+jβjβ)ejβr=β2(ejβrr)

Substitute R for ejβrr.

2R=β2R

As 2R=β2R, rewrite Equation (2).

2A=β2A

Therefore, Equation (1) is satisfied.

Find the electric scalar potential using Lorentz gauge expression.

V=1μoεoAdt

Substitute μo4πr[cos(θ)arsin(θ)aθ]ejωtejβr for A.

V=1μoεo[μo4πr[cos(θ)arsin(θ)aθ]ejωtejβr]dt=1jωμoεo[μo4πr[cos(θ)arsin(θ)aθ]ejωtejβr]=1jωμoεor[μo4πrejωtejβr]=1j4πωεo(jβr1r2)ejβrejωtcos(θ)

Simplify the expression.

V=cos(θ)j4πωεor(jβ+1r)ej(ωtβr)

Conclusion:

Thus, the given magnetic vector potential satisfies the expression 2Aμε2At2=μJ in free space and the electric scalar potential is cosθj4πωεor(jβ+1r)ej(ωtβr)_.

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