1. The potential distribution in free space is given by (a) Does V satisfy Laplace's equation?...
With steps 5) Does the function f(xy) -x+ y satisfy the two dimensional Laplace's equation? Does the function g(x,y)-x2-y2 ? Sketch g(x,y) roughly. And then calculate the gradient of g(x,y) at points (x,y)- (0,1), (1,0), (0, -1) and (-1,0) and indicate by little arrows the directions in which these gradient vectors point.
Thank you. 5. Find the solution u(x, y) of Laplace's equation in the rectangle 0<<a, 0<y<b that satisfies the boundary conditions u(0, y = 0, u(a, y) = 0,uy 3,0) = 0, 2,b) = g(1), where J2 0<x<a/2 g(x) = - 0<a/2 <<a
Problem 4. (25 points) Find the solution to the 2-dimensional Laplace's equation OLY + = 0 inside the square 0<x<1 0 <y <1 subject to the boundary conditions V(x,0) = 0 = V(x, 1) V(0,y) = 0 V(1,y) = 2 sin (31 y)
2. A region of space has a potential distribution that can be written as V(x, y, z) = -14xyz + 142 Volts, where x, y, and z are given in meters. a. (7 points) How much work is required to place a +10 uC charge at coordinates (x,y,z) = (10 m, 10 m, 10 m)? b. (7 points) What are the x-, y, and z-components of the electric field at coordinates (x,y,z) = (10 m, 10 m, 10 m)?
The equation of electric potential in space is given by: V(x,y,z) = 2xy/x 1. Calculate the electric potential at point (x = 1, y = -2, z = 3) in space. 2. Find the electric field E vector as a function of x, y, z. 3. Calculate the electric field at point (x = 1, y = -2, z = 3) in space.
V = 3. The potential in a region of space due to a charge distribution is given by the expression ax?z + bxy - cz? where a = -9.00 V/m3, b = 9.00 V/m², and c = 6.00 V/m2. What is the electric field vector at the point (0, -9.00, -8.00) m? Express your answer in vector form.
In a certain region of space, the electric potential is given by V = y^2 + 21xy - 11xyz. Determine the electric felid vector, E, in this region in terms of x, y, and z.
5) In free space, D 2ya,+4xya, - az mC/m2. Find the total charge stored in the region 1 <x < 2,1<y< 2, -1<z<4.
2. Solve for the bounded solution of Laplace's equation v2T=0 in the UHP: [2] < 0, y > 0 with the following boundary conditions given on y = 0: T(x,0) = {A on x < l1, B on li < x < l2,C on x > la} A, B, C are real constants.
1. (a) Derive the solution u(x, y) of Laplace's equation in the rectangle 0 < x <a, 0 <y <b, that satisfies the boundary conditions u(0,y) = 0, u(a, y) = 0, u(x,0) = 0, u(x,b) = g(x), 0 0 0 < a. (b) Find the solution if a = 4, b = 2, and g(x) = 0 <r <a/2, a-r, a/2 < x <a.