MT1A Spring 2022 Dr Ray Kwok 35 pts a What is the electric potential of a uniformly charged ring of radius R at a distance z away from the center as shown b Using the result from part a derive the electric potential of a charged disc with non uniform surface charge density r2 at a distance z away from the center as shown c From this electric potential write the electric field vector of the charged disc at this location z using EE140 1 z R z R VE 2222RzRrRkVrRdkrdqkdVo 32222022222 122 122 32 122220223022202222626332423244422222zRzzRVrzzruzuVuzuduuzuuduzuVrzurzdrrrzdrrrrzdrrdVVrdrrodiscRoodiscooodiscRoRoRoringdisc 222222222222322222 6246 622 zRzRzzzzRzzzRRzzzEzRzzRzzVEooo EE140 MT1A Spring 2022 Dr Ray Kwok 2 30 pts Consider a one dimensional problem An infinite line charge with uniform charge density is placed along the vertical axis and passing through x u as shown There are 2 parallel infinite ground planes at x d and d Write the net electric field at the origin o d n u o r g From Gauss s Law E field due to an infinite line of charge with uniform charge density is d n u o r g d x u 0 d 2 2 This is an infinite series It s like going into a dressing room with parallel mirrors 8d u 6d u u 2d u 4d u 2d u o 2 1 1 1 2 4 6 8 10 x 4d u 6d u 8d u 2 1 1 1 2 4 6 8 1 1 1 1 1 2 1 2 1 1 1 2 2 2 4 4 6 6 8 8 1 1 1 2 2 2 4 2 2 6 2 2 8 2 2 2 2 1 1 2 1 2 1 2 1 2 2 2 MT1A Spring 2022 Dr Ray Kwok 35 pts a Find the displacement vector D at a distance z away from an infinitely large uniformly charged plate on the xy plane with a negligible thickness as shown b Apply this result to find the electric field of a thick charged slab with a relative dielectric constant of 3 deposited with a non uniform charge density z2 as shown in figure To the left of the slab z 0 the medium has a relative dielectric of 2 To the right of the slab z t the medium has a relative dielectric of 4 as shown in the diagram Find the magnitude and direction of the electric field everywhere Sketch E z 2 2 pointing away from the charged plane For the block For z t z For z 0 same D symmetry Only because D is independent of z Part b For 0 z t we have 2 contributions why r 3 z r 4 z z 0 z t z t EE140 3 r 2 E t3 1 9 1 12 1 9 1 6 D ztDEztDtdzzDdzzdzdDoot 124 332 333022 dzAdzA ztDEztDoo 62 333 ootzztzzDEtzzztzDdzzdzzD 92 332 3333333333202
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