Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN: 9780190698614
Author: Sadiku, Matthew N. O.
Publisher: Oxford University Press
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Chapter 6, Problem 10P

(a)

To determine

State whether the given potential satisfies Laplace’s equation or not.

(a)

Expert Solution
Check Mark

Answer to Problem 10P

No, the given potential does not satisfy Laplace’s equation.

Explanation of Solution

Calculation:

Consider the given potential expression.

V1=3xyz+yz2        (1)

Write the expression for V2.

V2=V1x2+V1y2+V1z2        (2)

Substitute Equation (1) in (2).

V2=(3xyz+yz2)x2+(3xyz+yz2)y2+(3xyz+yz2)z2=(3yz)x+(3xz+1)y+(3xy2z)z=0+02

V2=2        (3)

Write the expression for the Laplace’s equation.

V2=0        (4)

Equation (2) is not equal to Laplace’s equation in Equation (4). Hence, the given potential equation does not satisfy the Laplace’s equation.

Conclusion:

No, the given potential does not satisfy Laplace’s equation.

(b)

To determine

State whether the given potential satisfies Laplace’s equation or not.

(b)

Expert Solution
Check Mark

Answer to Problem 10P

Yes, the given potential satisfies Laplace’s equation.

Explanation of Solution

Calculation:

Consider the given potential expression.

V2=10sinϕρ        (5)

Write the expression for the Laplace’s equation in cylindrical coordinates.

1ρρ(ρV2ρ)+1ρ22V2ϕ2+2V2z2=0        (6)

Substitute Equation (5) in (6).

1ρρ(ρ(10sinϕρ)ρ)+1ρ22(10sinϕρ)ϕ2+2(10sinϕρ)z2=01ρρ(ρ(10sinϕρ2))+1ρ2(10cosϕρ)ϕ+0=01ρρ(10sinϕρ)+1ρ2(10cosϕρ)ϕ+0=01ρ(10sinϕρ2)+1ρ2(10sinϕρ)=0

Simplify the equation as follows:

10sinϕρ310sinϕρ3=00=0

Hence, the given potential equation satisfies the Laplace’s equation.

Conclusion:

Yes, the given potential satisfies Laplace’s equation.

(c)

To determine

State whether the given potential satisfies Laplace’s equation or not.

(c)

Expert Solution
Check Mark

Answer to Problem 10P

No, the given potential does not satisfy Laplace’s equation.

Explanation of Solution

Calculation:

Consider the given potential expression.

V3=5sinϕr        (7)

Write the expression for the Laplace’s equation in spherical coordinates.

1r2r(r2V3r)+1r2sinθθ(sinθV3θ)+1r2sin2θ2V3ϕ2=0        (8)

Substitute Equation (7) in (8).

1r2r(r2(5sinϕr)r)+1r2sinθθ(sinθ(5sinϕr)θ)+1r2sin2θ2(5sinϕr)ϕ2=01r2r(5sinϕ)+0+1r2sin2θ(5cosϕr)ϕ=00+0+1r2sin2θ(5sinϕr)=05sinϕr3sin2θ0

Hence, the given potential equation does not satisfy the Laplace’s equation.

Conclusion:

No, the given potential does not satisfy Laplace’s equation.

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Elements Of Electromagnetics

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