Fa08-MT2-SunRev1-Soln

Fa08-MT2-SunRev1-Soln - Practice Midterm 2 Solution 1...

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Practice Midterm 2 Solution 1 Diamond Circuit a) First let’s calculate electric field due to two cylinder without dielectric. Uni- formly charge inner cylinder with total charge + Q and uniformly charge outer cylinder with total charge - Q . By symmetry around ˆ z -axis, we know electric field will be pointing in ˆ r direction. That is, vector E = E r ˆ r . Drawing gaussian pill-vox around cylinder with radius r , we can use gauss’s law, contintegraldisplay vector E · vector A = Q enc ǫ 0 (1) integraldisplay L 0 integraldisplay 2 π 0 E r ˆ r · ˆ rrdφdz = + Q ǫ 0 (2) integraldisplay L 0 integraldisplay 2 π 0 E r rdφdz = Q ǫ 0 (3) 2 πLE r r = Q ǫ 0 (4) E r = Q 2 πǫ 0 Lr (5) vector E = Q 2 πǫ 0 Lr ˆ r (6) We can find potential difference between two cylinder by using equation Δ V = - integraltext vector E · d vector l . Δ V = - integraldisplay vector E · d vector l (7) V b - V a = - integraldisplay b a Q 2 πǫ 0 Lr ˆ r · ˆ rdr (8) V b - V a = - integraldisplay b a Q 2 πǫ 0 Lr dr (9) V b - V a = Q 2 πǫ 0 L ln a b (10) Since we are interested in potential drop V a - V b = V , V = Q 2 πǫ 0 L ln b a (11) 1
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Definition of capacitance is C = Q V thus, C 0 = 2 πǫ 0 L ln b a (12) (You should have above expression in your equation sheet). But since there is dieletric between two cylinder C = KC 0 = K 2 πǫ 0 L ln b a (13) b) How to simplify capacitor is shown in Figure 1. Capacitors in series can be simplified by using relationship 1 C series = summationdisplay i 1 C i (14)
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