Question

5. Consider a long rectangular "gutter" of length a in the x direction and infi- nite height in the y direction. The gutter is infinitely long in the z direction, so the potential V inside the gutter only depends on x andy. The left (x-0,y), and right (r- a,y) sides of the gutter are grounded so that the potential V(x,y) is zero on those surfaces. The bottom surface of the gutter is kept fixed at a potential given by V(r,y-0)- Vo(r), where Vo(x) is an arbitrary function of x. (a) Derive a general solution for the potential V(x, y) inside the gutter. Clearly indicate any boundary conditions that you apply to reduce the full gen- eral solution of Laplace's equation to the particular solution for this prob- lem. You should show all non-trivial steps. Write down an integral expression that shows you how to calculate all of the unknown constants in your solution for an arbitrary potential Vo(x). (b) Now, assume the actual potential on the bottom of the gutter is is given by 27tx Vo(x) = 4 sin ( 2 ) +3sin Solve for the coefficients of the potential from part (a).

Answer 1

Given that V(x, y, z) = X(x) Y(y) Z(z) 1/x d^2 x/dx^2 + 1/r d^2 y/dy^2 + 1/z d^2 z/dz^2 = 0 1/x d^2 x/dx^2 = c_1: 1/r d^2 r/dy^2 = c_2 1/2 d^2 z/dz^2 = c_3 c_1 is + ve, c_2 and c_3 are -ve c_2 = -k^2, c_3 = -L^2, c_1 = k^2 + l^2 X(x) = Ae^squareroot (k^2 + x^2)x + Be^squareroot (k^2 + l^2) x As x rightarrow a X(x) = 0 = > A = 0 The boundary condition V = 0 at y = 0 and y = a z = 0 and z = downarrow k = r pi/a, L = m pi/b V(x, y, z) = (e^-pi squareroot (n/a)^2 + (m/b)^2 x sin(n pi y/a) + sin(m pi z/b)) \r\n The general solution is: V(x, y, z) = sigma^infinity_n = 1 sigma^infinity_m = 1 e^-pi squareroot (n/a)^2 + (m/b)^2 x sin(n pi y/a) sin() V(0, y, z) = sigma^infinity