(a)
State whether the given potential satisfies Laplace’s equation or not.
(a)
Answer to Problem 10P
No, the given potential does not satisfy Laplace’s equation.
Explanation of Solution
Calculation:
Consider the given potential expression.
Write the expression for
Substitute Equation (1) in (2).
Write the expression for the Laplace’s equation.
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)
State whether the given potential satisfies Laplace’s equation or not.
(b)
Answer to Problem 10P
Yes, the given potential satisfies Laplace’s equation.
Explanation of Solution
Calculation:
Consider the given potential expression.
Write the expression for the Laplace’s equation in cylindrical coordinates.
Substitute Equation (5) in (6).
Simplify the equation as follows:
Hence, the given potential equation satisfies the Laplace’s equation.
Conclusion:
Yes, the given potential satisfies Laplace’s equation.
(c)
State whether the given potential satisfies Laplace’s equation or not.
(c)
Answer to Problem 10P
No, the given potential does not satisfy Laplace’s equation.
Explanation of Solution
Calculation:
Consider the given potential expression.
Write the expression for the Laplace’s equation in spherical coordinates.
Substitute Equation (7) in (8).
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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Chapter 6 Solutions
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