Question

LE 4) (Ungraded) In Cartesian coordinates, the curl of a vector field Air) is defined as Use the definition of electric potential to find the potential difference between the origin and r = x + y + 27, V(r) - V(O) = - Ed. As the line integral is independent of path, choose whatever path you find to be con- vertient Taking V(0) = 0, what is V(r)? Finally, confirm that taking the gradient of the potential recovers our original electric field, E=- VA 04, 04 X + OA -- ) + DA, 24, - ay chy 2. In Math 32B you learn about Stokes's theorem, which relates the surface integral of a vector field's curl to the line integral of the vector field around the surface's boundary, $4.d = [(5xA).da. Equipotential surfaces (positive in orange and negative in blue) are shown below, The orientation of the surface is determined by direction of the boundary traversal through the right-hand rule, as shown below. Flowing through all space is an electric field E[.z, , ] = 0,2k+12ỷ+T5). Show that the curl of the electric field vanishes, F x E = (). Then, by Stokes' theoremn, the line integral around a closed path zero, fe E dI = 0, and we say the clectric field is conservative. Morcover, this implics that line integral from a to b ia independent of the path taken. To show this, given two separate patha from a to b, simply form a closed loop from the two paths (one going in reverse) around which the line integral alust vanish.

Answer 1

Po C 4 Giren: FOXY, Z) = xyz + 8x2 ;+ Kx42 DX Ez ☺ z Oy 2 By 22 242 Laz ie O S z (2-x) = (y - 049 + (Q2-0272 TXE 20 By stoke's Theorem, fande = f (8XA), da pe Edda (Cox Eids zo. Te lo Edizo VW - VO) -- Med Volzo 4 Venz? Ez-DV Solution: +9 By definition of electric potential, +9 vz- dr Edr 49. -9 So giren cube of length, a, the equipotential given by +9 thire & -tive charges. How, from those surface , we can assume Нng, choляд corners os So, when calculating potential b from electic field on equipontential lines, the potential cancel along length a. This for along any length of cube, Ezo. Iz Vozo then Vnzo & Ez-DV co. surfaces are a cube. lie. potentia zo.