A vector field is given by F = y'e*i - x*yj - xtan-'yk use the divergence (Gauss") theorem, to calculate the flux of F through the surface S of the solid D bounded by the circular Cylinder x? + y? = 4, and the planes z = 0 and z = 3. z-3 %3D
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- Discuss the possibility of choosing Gaussian surface other than the sphere.For instance what if you choose the cylindrical surface as a Gaussian surface, is it possible to use Gauss’s law in order to calculate the electric fieldof a charged sphere? Explain your answer in detail.Calculate the flux of the given vector field by evaluating the line integral directly alongthe given curve for the below parts:(a) The vector field is ⃗ F = (x − y)⃗i + x⃗j. The curve is the circle x^2 + y^2 = 1in the xy-plane. Use the parameterization x = cos t and y = sin t.(b) The vector field is ⃗ F = (x − 1)⃗i + y⃗j. The curve is a circle of radius 3centered at (1, 1). The parametric form of this circle is⃗r = (1 + 3 cos t)⃗i + (1 + 3 sin t)⃗j, 0 ≤ t ≤ 2π(c) The vector field is ⃗F = x⃗i + y⃗j. The curve is the line segment from thepoint (0, 1) to the point (1, 3).Explain the three gaussian surfaces
- A vector field is pointed along the z-axis, v=ax2+y2z . (a) Find the flux of the vector field through a rectangle in the xy-plane between axb and cyd. (b) Do the same through a rectangle in the yz-plane between azb and cyd. (Leave your answer as an integral.)Repeat the previous problem, given that the circular area is (a) in the yz-plane and (b) 45( above the xy-plane.A cubical gaussian surface is bisected by a large sheet of charge, parallel to its top and bottom faces. No other charges are nearby. (i) Over how many of the cubes faces is the electric field zero? (a) 0 (b) 2 (c) 4 (d) 6 (ii) Through how many of the cubes faces is the electric flux zero? Choose from the same possibilities as in part (i).
- A fellow student calculated the flux through the square for the system in the preceding problem and got 0. What went wrong?Find the electric flux through the plane surface shown in the figure below if θ = 63.6°, E = 339 N/C, and d = 5.40 cm. The electric field is uniform over the entire area of the surface.The Electric Flux and Gauss’ Law A flat sheet is in the shape of a rectangle with sides of lengths 0.400 m and 0.600 m. The sheet is immersed in a uniform electric field of magnitude 75.0 N/C that is directed at 20° from the plane of the sheet. Find the magnitude of the electric flux through the sheet.
- A non-uniform thin rod is bent into an arc of radius R. The linear charge density λ of the roddepends on θ and is given byλ =λ0/cos θwhere λ0 is a positive constant. The arc extends from θ =π/4 to θ =3π/4as shown a)Sketch the direction of the resultant electric field at the origin.b) Calculate the magnitude of the electric field E->.What is the correct answer: Gauss's law a. relates the total electric flux through a closed surface with the net electric charge enclosed within the surface. b. implies that in static situations any excess charge on an insulator must lie on its surface. c. is very useful for charge configurations with asymmetry. d. all of the above.Problem 2: A closed hollow cylinder (i.e., with capped ends) is situated in an electric field given by E(u) = E0(u5i + 7j + 22k). The cylinder’s axis is on the x-axis with its center at the origin. The cylinder’s height is L and its radius is R. Here u = x/x0 is a dimensionless variable, where x0 sets the scale of the field. Refer to the figure. Part (a) Integrate to find an expression for the total electric flux through the cylinder in terms of defined quantities and enter the expression. Part (b) For L = 8.7 m, R = 0.25 m, E0 = 4.5 V/m, and x0 = 1 m, find the value of the electric flux, in units of volt•meter, through the cylinder. Part (c) If the electric field is E(u) = E0(323u2i + 42j + 415k), enter an expression for the total flux in terms of defined quantities.