Consider, in the xy-plane, the upper half disk of radius R, with temperature u governed by...

Consider, in the xy-plane, the upper half disk of radius R, with temperature u governed by Laplace's equation, and with zero-Dirichlet B.C. on the (bottom) flat part of the disk, and Neumann B.C. ur=f(theta), on its (top) curved boundary.

(a)Use the Method of Separation of Variables to completely derive the solution of the BVP. Show all details of the procedure, and of your work. Do not start with with the eigenfunctions, the are to be derived. The final solution must be summarized and circled at the end of your work, showing the final complete expression for u(r,theta) and any coefficents and necessary conditions for f(theta).

(b) Obtain the complete simplified solution of the BVP above, with R=2 and f(theta) = 100.

Expert Solution

Colby Messinger answered 2 years ago

Consider a charged disk of radius R on the x-z plane with its centre at the...

Consider a charged disk of radius R on the x-z plane with its centre at the origin. The disk has a positive charge density σ. (a) Find, from first principles, an expression for the electric field of this disk at point P (0,yP,0) on the axis of the disk. (b) A second identical charged disk is now placed at a distance d parallel to the first, with its center at (0,d,0). Find the net electric field due to both disks...

a) Let D be the disk of radius 4 in the xy-plane centered at the origin....

a) Let D be the disk of radius 4 in the xy-plane centered at the origin. Find the biggest and the smallest values of the function f(x, y) = x 2 + y 2 + 2x − 4y on D. b) Let R be the triangle in the xy-plane with vertices at (0, 0),(10, 0) and (0, 20) (R includes the sides as well as the inside of the triangle). Find the biggest and the smallest values of the function...

A thin, circular disk of radius R = 30 cm is oriented in the yz-plane with...

A thin, circular disk of radius R = 30 cm is oriented in the yz-plane with its center as the origin. The disk carries a total charge Q = +3 μC distributed uniformly over its surface. Calculate the magnitude of the electric field due to the disk at the point x = 15 cm along the x-axis.

1. Consider the spiral S in the xy-plane given in polar form by r = e...

1. Consider the spiral S in the xy-plane given in polar form by r = e ^θ . (a) (5 points) Find an arc length parametrization for S for which the reference point corresponds to θ = 0. (b) (5 points) Compute the curvature of S at the point where θ = π.

Consider a thin uniform disk of mass M and radius R. A mass m is located...

Consider a thin uniform disk of mass M and radius R. A mass m is located along the axis of the disk at a distance z from the center of the disk. The gravitational force on the mass m (in terms of m, M, R, G, and z) is

Consider the solid that lies above the rectangle (in the xy-plane) R=[−2,2]×[0,2], and below the surface...

Consider the solid that lies above the rectangle (in the xy-plane) R=[−2,2]×[0,2], and below the surface z=x2−4y+8. (A) Estimate the volume by dividing R into 4 rectangles of equal size, each twice as wide as high, and choosing the sample points to result in the largest possible Riemann sum. Riemann sum =? (B) Estimate the volume by dividing R into 4 rectangles of equal size, each twice as wide as high, and choosing the sample points to result in the...

Acircular ring of radius ?lies in the ??plane and is centered on the origin. The half...

Acircular ring of radius ?lies in the ??plane and is centered on the origin. The half on the positive ?side is uniformly charged with a charge +?while the half on the negative ?side is uniformly charged with a total charge −?. a..Draw a diagram of the charge distribution and without doing any math, determinethe direction of the total electric field at the origin . b.Calculate the ?component of the electric field at the origin by integrating the charged ring. i.Draw...

A uniform disk with radius 0.310 m and mass 30.0 kg rotates in a horizontal plane...

A uniform disk with radius 0.310 m and mass 30.0 kg rotates in a horizontal plane on a frictionless vertical axle that passes through the center of the disk. The angle through which the disk has turned varies with time according to ?(t)=( 1.10 rad/s)t+( 6.90 rad/s2 )t2 What is the resultant linear acceleration of a point on the rim of the disk at the instant when the disk has turned through 0.100 rev ?

A solid disk of mass M and radius R is rotating on the vertical axle with...

A solid disk of mass M and radius R is rotating on the vertical axle with angular speed w. Another disk of mass M/2 and radius R, initally not rotating, falls coaxially on the disk and sticks. The rotational velocity of this system after collision is: w w/2 2w/3 3w/2 2w

Find the moment of inertia of a circular disk of radius R and mass M that...

Find the moment of inertia of a circular disk of radius R and mass M that rotates on an axis passing through its center. [Answer: ½ MR2] Step 1: Pictorial representation: Sketch a neat picture to represent the situation. Step 2: Physical representation: 1) Cut the disk into many small rings as it has the circular symmetry. 2) Set up your coordinate system and choose its origin at the pivot point (or the axle location) for convenience. Then choose a...

Question