25 6 example a 1m2 d 1 m a c 0 885 10 12 106

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Unformatted text preview: eometry of device ++++++++++++++++ E d --------------------- PHY2049: Chapter 25 6 Example A = 1m2, d = 1 μm A C = ε 0 = 8.85 × 10-12 × 106 = 8.85 × 10-6 F d C = 8.85 μ F Largish, but somewhat typical value PHY2049: Chapter 25 7 Cylindrical Capacitor Inner = a, outer = b, length = L Gauss’ law: Using λ=q/L + V = ∫ E ds − V = −∫ a b q1 E= 2π Lε 0 r (ds = −dr) b - - + + + - - a + - + - q ln ( b / a ) Edr = 2π Lε 0 C 2π = ε0 L ln ( b / a ) - - + + - + - - Capacitance per unit length, e.g. “coaxial” cable (RF frequencies) PHY2049: Chapter 25 8 Special Case for Cylinder Outer shell “very close” to inner shell: b − a = d (d small) Use ln(1+x) ≅ x (for x small) ⎛b⎞ ⎛a+d⎞ d ln ⎜ ⎟ = ln ⎜ ⎟ a⎠ a⎠a ⎝ ⎝ Asurface 2π L 2π aL ≅ ε0 = ε0 C = ε0 ln ( b / a ) d d A Just like parallel plate capacitor: C = ε 0 d Always true if surfaces are close together PHY2049: Chapter 25 9 Spherical Capacitor Inner radius = a, outer radius = b Coulomb’s law: + V = ∫ E ds − V = −∫ a b E= q b - 1 4πε 0 r 2 + - - a + q b−a Edr = 4πε 0 ab - + + - + - (ds = −dr) - - + + - + - - 4π ab C = ε0 b−a PHY2049: Chapter 25 10 Two Special Cases Isolated sphere: corresponds to b = ∞ 4π ab 4π a = ε0 → ε 0 4π a C = ε0 b−a 1− a / b Outer shell “very close” to inner shell: b − a = d (d small) Asurface 4π ab 4π a ≅ ε0 = ε0 C = ε0 b−a d d A C = ε0 d 2 Again, just like parallel plate: PHY2049: Chapter 25 11 C...
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This note was uploaded on 07/31/2011 for the course PHY 2049 taught by Professor Any during the Summer '08 term at University of Florida.

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