Concept explainers
For the parallel-plate capacitor shown in Figure 6. 3, find the potential field in the interior if the upper plate (at z = d) is raised to potential V0, while the lower plate (at z = 0) is grounded. Do this by solving Laplace' s equation separately in each of the two dielectrics. These solutions, as well as the electric flux density, must be continuous across the dielectric interface. Take the interface to he at z = b.
The potential field between plates
Answer to Problem 6.31P
The values are
Explanation of Solution
Calculation:
The Poisson's equation is defined for
The Poisson's equation (generalization of Laplace equation) is written as
Here,
Integrate the equation (1) with respect to
The Poisson's equation is defined for the region
The Poisson's equation is written as
Integrate the equation (3) with respect to
Substitute
Simplified the equation (5) as
Equation (4) is multiplied with
If equation (6) and equation (7) is equal, then it is written as
Simplified the equation (8) as
Substitute
Integrate the equation (2) with respect to
Integrate the equation (9) with respect to
Substitute
Equation (12) is simplified as
Substitute
Substitute
If equation (13) and equation (14) is equal, then it is written as
Substitute
Substitute
Substitute
The potential difference
Substitute
Substitute
Conclusion:
Therefore, the final voltage
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Chapter 6 Solutions
Engineering Electromagnetics
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