let F be a vector field whose curl and divergence at the origin are curl(F)(0, 0, 0) = (2, – 1, 4) , div(F)(0, 0, 0) = -2 Estimate the flux of F through the box of side 0.5 in Figure . Does the result depend on how the box is oriented relative to the coordinate axes? 0.5 FIGURE
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- 3 Consider a uniform source of particles inside a box with dx=8 cm , dy=4 cm, and dz=2 cm, centered on the origin. Use hand calculation to model each of the sources (3D, 2D-rectangular x-y, 1D linear x, point at origin) and compare the total flux at distances of 1, 2, 64, and 256 cm above from the origin (in the z direction).c) For the same cylindrical shell as in the previous problem, draw and label a Gaussian surface and use Gauss's Law to find the radial electric field in the region r > R2. You may take the positive direction as outward. 0 E (r > R2) =Below is a graph of electric flux as a function of sphere radius r. What is the symbolic value of r at point P where the two sections meet?
- Problem 1 (25%) An infinite plane slab of thickness 2d (the slab is located between z=-d and z=d) carries a uniform volume charge density p. Using the Gauss's law, find the electric field as a function of z inside and outside the slab. Plot schematically the dependence of the electric field as a function of z. 2d $000ric Fields due to Point Charges An uneven dipole is located along the x-axis and centered on the origin as drawn. Its positive end has twice as much 2a 1 charge as its negative end. Each charge is located a distance a away from the origin along the x-axis. In terms of k, q, and a, find the electric field vector only at а POINT 1 and POINT 2. Your answers should be simplified 2 numerically until they are in the form: E = (0.123, -2.345), for example. -2a +29 2a1A On the diagram below, draw the normal vectors for all six faces of the cube (have the tail of the vector placed at the dots, 1 mark). What is the surface area vector for the top (3) and the bottom (6) faces of the cube in terms of î, ĵ, k.I B)| Write expressions of the electric field Ē along each of the faces of the cube. E, (a, y,z) = Ē2(x, a, z) = Ē3(x, y, a) = E, (0, y, z) = Es(x, 0, z) = Ē,(x, y,0) = C) Calculate the value of ¤g = S Ē -dà o on face 3 of the cube. Show all your steps.