ps04sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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PS04-1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2010 Problem Set 4 Solution Problem 1: Experiment: Expt. 2: Faraday Ice Pail Capacitance of our Experimental Set-Up Part 1 Consider two nested cylindrical conductors of height h and radii a & b respectively. A charge + Q is evenly distributed on the outer surface of the pail (the inner cylinder), -Q on the inner surface of the shield (the outer cylinder). (a) Calculate the electric field between the two cylinders ( a < r < b ). For this we use Gauss’s Law, with a Gaussian cylinder of radius r , height l 00 0 1 2( ) 2 inside arb Q QQ dr l E lE r hr h π ε επ << ⋅= = = = ∫∫ EA r r ± (b) Calculate the potential difference between the two cylinders: The potential difference between the outer shell and the inner cylinder is 0 () l n l n 22 2 a a b b QQQ b VV aV b d r r rh h h a π ε πε ⎛⎞ ′′ Δ= = = = ⎜⎟ ⎝⎠ (c) Calculate the capacitance of this system, C = Q/ Δ V 0 2 || ln ln 2 o h C Qb b V ha a == = Δ
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PS04-2 (d) Numerically evaluate the capacitance for your experimental setup, given: h 15 cm, a 4.75 cm and b 7.25 cm 9- 1 2 11 5 c m 20 pF 7.25 cm 29 10 m F ln ln 4.75 cm o h C b a πε == ⋅× ⎛⎞ ⎜⎟ ⎝⎠ e) Find the electric field energy density at any point between the conducting cylinders. How much energy resides in a cylindrical shell between the conductors of radius r (with arb << ), height h , thickness dr , and volume 2 rhdr π ? Integrate your expression to find the total energy stored in the capacitor and compare your result with that obtained using 2 (1/ 2) ( ) E UC V Δ = . The total energy stored in the capacitor is 2 2 00 0 22 2 E Q uE rh εε Then 2 2 0 1 2 4 E QQ d r dU u dV hr επ = Integrating we find that ln( / ) 44 bb aa Qd r Q Ud U b a h = ∫∫ .
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This note was uploaded on 04/20/2010 for the course PHYSICS 8.02 taught by Professor Hughes during the Spring '08 term at MIT.

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ps04sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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