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homework_7 - C ab for a 6 = b are called at least by some...

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ELECTRODYNAMICS PROBLEM SET 7 due March 16 th , before class Problem 1: Dielectric fun Two concentric conducting spheres of radii a and b carry charges ± Q . The empty space between them is half-filled by a hemispherical shell of dielectric (with dielectric constant ). Find the electric field everywhere between the spheres. Problem 2: Straight from Jackson A very long, right circular, cylindrical shell of dielectric constant and inner and outer radii a and b , respectively, is placed in a previously uniform electric field E 0 with its axis perpendicular to the field. The medium inside the cylinder is air 1. a) Determine the filed in all three regions neglecting edge effects. b) Sketch the lines of force for the case b = 2 a . Problem 3: Capacitances Consider N conducting bodies in space. Since Maxwell’s equations are linear, their charges and potentials are related linearly Q a = N X b =1 C ab φ b . (1) The constants C aa are called capacitances and the non-diagonal constants
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Unformatted text preview: C ab for a 6 = b are called, at least by some, electrostatic induction coefficients. They depend only on the shape of the conductors (and dielectrics, if present). a) Argue that C ab = C ba and that C aa > 0. Hint: write down the energy in terms of the C ab . b) In the gaussian system, what are the units for C ab ? What is C for a sphere? c) The Earth is a good conductor and approximately spherical. What is the capacitance of the Earth in “practical” units (microfarads)? If we remove all the electrons out of a gram or so of material on the Earth, what would be the potential of the Earth? What would be the electric field on the Earth’s surface (in volts/cm)? Hint: you can find conversion factors for electromagnetic quantities on the back cover of textbooks. Google can probably do it too). Hint 2: I’m looking only for rough numbers here....
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