+ X C Relaunch to update E 1 / 5 99% + Problem 1. As observed in lectures, one of the most important types of vector fields in physics and engineering is the electric field E. If we place an electric charge at the point xo = (To, yo, 20) in space, it generates a vector force field E(x, y, z) given by: X - TO y - yo 2 - 20 E(x, y, z) = ATEO Tac - 20|3' |2 - 20|3' |20 - 2003 where q is the electric charge, and zo is a constant representing the permittivity of air, and lac | represents the magnitude of the vector x, i,e., |2c | = Vx2+ 12+ 22. (sometimes, the magnitude is denoted by ||.I|, that is, llac || = Vx2 + 2 + 22). a) Prove that E is a conservative force field with the potential energy: U(x, y, z) = ATEO ac - 20 in the sense E = -VU. b) Two electric charges, q and -q, are located at the points (1, 0,0) and (-1, 0, 0), respectively. Find the electric field E and the associated potential energy U at an arbitrary point (x, y, z) that is not (1, 0, 0) or (-1, 0, 0). The electric field and potential energy follow the principle of superposition, where they result from the addition of two fields and potentials. Use the following code to visualize the electric field and contour lines of the potential in the xy-plane. Attach the image to your solution: x=-2:0.1:2; y=x' ; u= (x-1) . / ((x-1) . -2+y. 2) . ~(3/2) -(x+1) . /((x+1) . -2+y. -2) . -(3/2) ; v=y . / ( (x-1) . ~2+y. -2). ~(3/2)-y. /((x+1) . -2+y. -2). ~(3/2) ; streamslice (x , y , u, v, 0.5) axis tight equal hold on z=1. /sqrt ((x-1) . ~2+y. 2)-1./sqrt ((x+1) . ~2+y. -2) ; contour (x , y, z, [-2 -1 1 2], 'color' , 'r' ) p1 = nsidedpoly (1000, 'Center' , [1 0], 'Radius' , 0.12) ; p2 = nsidedpoly (1000, 'Center', [-1 0], 'Radius', 0. 12) ; plot (p1, 'FaceColor', 'b') plot (p2, 'FaceColor', 'r' ) You can observe that the stream lines are perpendicular to the contour curves which repre- sents equipotential curves. -1. C ENG 9:58 AM Sunny US 2023-11-20