Concept explainers
The solution of Laplace' equation.
Answer to Problem 6.43P
The potential field
Explanation of Solution
Calculation:
The Poisson's equation (generalization of Laplace equation) is defined for
The Poisson's equation is written as
Here,
Simplify equation (1) as,
Integrate equation (2) with respect to spherical coordinate
Substitute
Simplified the equation (4) as,
The Laplace's equation is defined for
The Laplace's equation is written as
Integrate the equation (5) with respect to spherical coordinate
Equation (6) is multiplied with
If equation (6) and equation (7) is equal, then it is written as
Simplified the equation (8) as,
Substitute
At boundary condition
Substitute
Simplify equation (12) as
Integrate the equation (3) with respect to
Substitute
Simplify equation (14) as
Integrate the equation (10) with respect to
Substitute
Equation (16) is simplified as
Substitute
Substitute
If equation (17) and equation (18) is equal, then it is written as
Substitute
Substitute
Substitute
The value of potential difference
Substitute
Substitute
Conclusion:
Therefore, the potential field
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Chapter 6 Solutions
Engineering Electromagnetics
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