Boundary Value Problems: A Square Pipe
An infinitely long, square pipe has three grounded sides at y=0, y=a, and x=0. The fourth side, at x=a, is maintained at a constant potential, V0. Find the potential inside the pipe.
QUESTIONS
(Score is number right minus number wrong.)
There must be charge on the surface to maintain the potential V0 at x=a. Therefore, the potential inside the pipe must satisfy Poisson's Equation.
True
False
Equations (4) and (5) both satisfy Laplace's Equation.
True
False
Equation (4) and Equation (5) are general solutions that can always be used in Cartesian coordinates for a square pipe held at some potential or combination of potentials.
True
False
Equation (4) or Equation (5) can only be used for a square pipe if the boundary conditions (potential) are sufficiently limited (i.e., not completely general).
True
False
Equation (5) should be used for the pipe in question because the boundary conditions are periodic (repeated) for y=0 and y=a.
True
False
To fit the periodic (repeated) boundary condition along the y-axis of the pipe, we should use Equation (4) rather than Equation (5) because the sine and cosine functions are naturally periodic.
True
False
It is NOT permissible to let some of the constant coefficients (A, B, C, D) in Equation (4) vanish because the resulting expression would no longer necessarily satisfy Laplace's Equation.
True
False
Equations (4) and (5) only satisfy Laplace's Equation for certain discrete choices of the parameter 'k'.
True
False
To fit boundary condition (10a), we should choose,
Equation (11a)
Equation (11b)
Equation (11c)
Equation (11d)
Equation (11e)
Equation (11f)
Equation (11g)
Equation (11h)
Equation (11i)
Equation (11j)
To fit boundary condition (10b), we should choose,
Equation (11a)
Equation (11b)
Equation (11c)
Equation (11d)
Equation (11e)
Equation (11f)
Equation (11g)
Equation (11h)
Equation (11i)
Equation (11j)
To fit boundary condition (10c), we should choose,
Equation (11a)
Equation (11b)
Equation (11c)
Equation (11d)
Equation (11e)
Equation (11f)
Equation (11g)
Equation (11h)
Equation (11i)
Equation (11j)
An expression that satisfies Laplace's Equation and the first three boundary conditions in Equation (10) is
Equation (12a)
Equation (12b)
Both
Neither
Equation (12b) satisfies Laplace's Equation because the Laplacian operator (del-squared) is linear. If each term in the sum is independently a solution, then a sum of solutions is also a solution.
True
False
We can find the unknown coefficients (A's) by multiplying both sides of Equation (12b) by a suitably chosen sine-function and using the orthogonality conditions summarized in Equations (7)-(9).
True
False
The coefficients (A's) in Equation (12b) are correctly given by,
Equation (13a)
Equation (13b)
Equation (13c)
Equation (13d)
Equation (13e)
Equation (13f)
Equation (13g)
EQUATIONS
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