331. F(x, y, z) = 2yi – 6zj + 3æk; S is a portion of paraboloid z = 4 – x2 – y? and is above the xy-plane.
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- = Let er be the unit radial vector field. Compute the outward flux of the vector field F er/r² through the ellipsoid 4x² + 6y² + 9z² = 36. [Hint: Because F is not defined at zero, you cannot use the divergence theorem on the bounded region inside of S. ]In free space, let D = 8xyz*ax + 6x²z*ãy + 16x?yz'a, pC/m?. a. Find the total electric flux passing through the rectangular surface z = 2, 0E1. Consider the vector field in spherical coordinates, F(r) = n (a) Find the closed line integral around path C that is a circle of radius s in the x-y plane and centered on the origin. $. F. dl (b) Find the surface integral of curl F over the hemispheric surface,H, enclosed by C: V X F. da (c) Find the surface integral of curl F over the circular disk area, D, enclosed by C. (d) Demonstrate that for both surfaces H and D that Stoke's theorem works.QUESTION 8 Calculate Curl D at the point specified if D=10xyz i+ 5x²zj + (2z³ - 5x²y) k, at P(-2,3,5).1. If I draw a Gaussian surface surrounding the nucleus of an Oxygen atom, with an atomic number of 8, what is the electric flux through that surface? If I increase the size of that surface to enclose the entire atom, i.e., include the electrons too, what is the electric flux through the surface? What is the electric flux if the Oxygen atom oxidizes to 0²-?Find the flux of F = xi - 2yj + zk across the portion of cylinder x² + z² = 9 in the first and forth octants. (3,-3,0) n X (0,0,3) (3,0,0) (0,3,0) y29 %3D 91 92 %3D E O The electric flux is shown through two Gaussian surfaces. In terms of q, what are charges q1 and q2? O q1 = 2q; q2 = q %3D O q1 = q; q2 = 2q %3D O q1 = 2q; q2 = -q %3D O q1 = 2q; q2 = -2q %3D O q1 = q/2; q2 = q/2 ||Q1. The volume charge density of a spherical shell with inner radius a and outer radius b is given as Pv = Po/R as shown in Figure 1. Determine E in all regions. (po is a positive constant) Pu Figure 1Sketch the diagram to illustrate the primitive vectors of a face centred cubic, FCC and state the primitive vector equations in the terms of Cartesian unit vectors. Calculate the volume of primitive unit cell.Calculate the flux of the vector field F(x, y, z) = 5i + 5j + zk through the closed circular cylinder of radius 4 centered about the z-axis for -6 ≤ z < 6, oriented away from the z-axis. Note: a closed cylinder has a top and a bottom. Flux = SS F.dĀ=6. By evaluating both sides of the equation, verify the divergence theorem using the field Ġ= (x² + y² + ²)(xi+yĵ+zk) and the region bounded by the sphere x² + y² + ² = 25.