1) (Laplace equation in two dimensions) (a) Consider complex coordinates z = x + i y and the 45 degree "cake slice" V = {z ? C | arg(z) ? [0, ?/4], |z| ? [0,a]} in the complex plane. Using complex methods, find solutions ? to the two-dimensional Laplace equation on V which satisfy the boundary conditions ?|_{arg z = 0} = 0 = ?|_{arg z = ?/4}. (b) From the solutions found in part (a), select the one which, in addition, satisfies the boundary condition ?|_{|z|=a} = h, where h : [0, ?/4] ? R is a given function. (c) Show that the most general solution to the two-dimensional Laplace equation in polar coordinates can be written as ?(r,?) = a0/2 + 5a0 ln r + 55555555555 555 (a_k r^k +
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Let Φ(θ, r) = Θ(θ)R(r) be the separation of variables. Show more…
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Consider the Laplace equation problem in the rectangle 0 < x < 3 and 0 < y < 4: u_{xx} + u_{yy} = 0 u(0, y) = 0; u(3, y) = f(y) u(x, 0) = 0; u(x, 4) = 0 If you were to solve this problem by separation of variables by writing u(x, y) = X(x)Y(y), what would be the solutions for X_{n} and Y_{n}? (a) X_{n} = sin (frac{npi x}{3}); Y_{n} = cos (frac{npi y}{4}) (b) X_{n} = sin (frac{npi x}{3}); Y_{n} = sin (frac{npi y}{4}) (c) X_{n} = - tanh (frac{3npi}{4}) cosh (frac{npi x}{4}) + sinh (frac{npi x}{4}); Y_{n} = sin (frac{npi y}{4}); (d) There is no solution (e) None of these (f) X_{n} = sinh (frac{npi x}{4}); Y_{n} = sin (frac{npi y}{4}) (g) X_{n} = sin (frac{npi x}{3}); Y_{n} = - tanh (frac{4npi}{3}) cosh (frac{npi y}{3}) + sinh (frac{npi y}{3}) (h) X_{n} = sin (frac{npi x}{3}); Y_{n} = sinh (frac{npi y}{3}) (i) X_{n} = tan (frac{4npi}{3}) cos (frac{npi x}{3}); Y_{n} = sin (frac{npi y}{4}) (j) X_{n} = tan (frac{4npi}{3}) cos (frac{npi x}{3}); Y_{n} = cos (frac{npi y}{4})
Adi S.
Consider a three-dimensional potential U which obeys the Laplace equation ∇²U = 0. In spherical polar coordinates (r, θ, ϕ), and assuming that the potential does not depend on ϕ, the Laplace equation can be written as 1/r² ∂/∂r (r² ∂U/∂r) + 1/(r² sin θ) ∂/∂θ (sin θ ∂U/∂θ) = 0. (i) We can solve the Laplace equation using a trial function U(r, θ) = R(r)Θ(θ). Show that 1/R d/dr (r² dR/dr) = k, and 1/Θ d/dμ ((1 - μ²) dΘ/dμ) + k = 0, where μ = cos θ and k is a constant. (ii) You may assume that the differential equation for Θ has valid solutions given by the Legendre polynomials Pℓ(μ) only when k = ℓ(ℓ + 1), where ℓ = 0, 1, 2, 3, ... are positive integers. Show that R ∝ rᴬ is a valid solution, and determine the two possible values of A in terms of the numbers ℓ. (iii) Suppose that the angular dependence of the potential on the surface of the unit sphere is given by Θ(θ) = 1/2 (3 cos² θ - 1). Using your result from (ii), find the potential both inside and outside the sphere (no sources) assuming that R(r = 1) = 1. Hints: Recall the second Legendre polynomial P₂(μ) = 1/2 (3μ² - 1), and ∫₋₁¹ dμ Pℓ(μ) Pₘ(μ) = { 2/(2m + 1) (l = m), 0 (l ≠ m).
Sri K.
The three-dimensional Laplace equation $\mathbb\quad \frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}+\frac{\partial^{2} f}{\partial z^{2}}=0$ is satisfied by steady-state temperature distributions $T=f(x, y, z)$ in space, by gravitational potentials, and by electrostatic potentials. The two-dimensional Laplace equation $$ \frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}=0 $$ obtained by dropping the $\partial^{2} f / \partial z^{2}$ term from the previous equation, describes potentials and steady-state temperature distributions in a plane. The plane may be treated as a thin slice of the solid perpendicular to the $z$ -axis. Show that each function in Exercises satisfies a Laplace equation. $$f(x, y)=\tan ^{-1} \frac{x}{y}$$
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