Check Gauss's theorem on a sphere of radius R using the following vector 1 i = 4Tr2 You'll get a surprising answer. Can you sce where the problem is?
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- Consider the special shape pictured in the diagram below. It is a cylinder, centered on the origin with its axis oriented along z, and it has been partially hollowed to leave two cone-shaped cavities at the top and bottom of the cylinder. The radius of the object is a, its height is 2a, and the solid part of the object (the shaded region that is visible in the rightmost panel of the illustration above, which shows a drawing of the cross-section of the object) has a uniform volume charge density of po. Assume that the object is spinning counter clockwise about its cylinder axis at an angular frequency of w. Which of the following operations is part of the calculation of the magnitude of the current density that is associated with the motion of the rotating object as a function of r (select all that apply)?Let C be the region between the circles x² + y² = 1 and x² + y² = 4. Let 2 2 2 F(x, y) = (²3³-²3V², 30³² Compute the circulation across C. Flux = y³, x³ - 3x²A cylinder of radius R has a charge density given by p = Po/r where po is a constant, for r R. Find the electric field both inside and outside the cylinder using Gauss' law. 1.
- A thin spherical shell of radius R carries a total charge Q. We define theorigin of our coordinate system to be in the center of the sphere enclosedby this shell.(a) Calculate the electric field due to the spherical shell just outside theshell in two ways: 1) by Gauss’s law and 2) by applying our knowledgeon the boundary conditions of electric fields. Do the two methodsagree?Find the electric field due to infinitely long cylindrical volume charge density (p,) and a cylindrical surface charge density (p,) for pA hollow sphere made from a non-conducting material is shown below in cross-section. The inner radius is R1, and the outer radius is R2. The material is charged uniformly -ρ.(a) Using Gauss’ Law, find an expression in terms R1, R2, and ρ of the magnitude(measured in N/C) and the direction (away from or towards the center) of the electricfield at a distance r from the center of the sphere, for values R1 < r < R2? (Do all thiswork symbolically - don't use the values of part (b) for this part.) (b) The inner radius is R1 = 1.00 cm, and the outer radius is R2 = 4.00 cm. Thematerial is charged uniformly ρ = -1.70 nC/m3. If the electric potential is 0V infinitely faraway, what is the electric potential at the outer surface (r = R2) of the sphere? Please show full work Thank you!Question 1: Gauss' Law: Electric field from surface charge For this problem assume constants are all in SI units that are not shown. An electric field given by E = (3z³+5)k. Calculate the flux through surface S, with side length a = 2, shown below if the surface is at the following heights: (a) z = 0 (b) z = -2 (c) z = 2 E 2.0 m 1.5 m a (d) The electric field in a region is given by E= a/(b + cx)î, where a = 500N m/C, b = 4.0m, and c 1.0. What is the net charge enclosed by the shaded volume shown below? 1.0 m a 425 y(a) State Gauss’ law. Using this law, obtain the expression for the electric field due to an infinitely long straight conductor of linear charge density X. (b) A wire AB of length L has linear charge density λ = kx, where x is measured from the end A of the wire. This wire is enclosed by a Gaussian hollow surface. Find the expression for the electric flux through this surface.