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 coecients. 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 eld on the Earths surface (in volts/cm)? Hint: you can nd conversion factors for electromagnetic quantities on the back cover of textbooks. Google can probably do it too). Hint 2: Im looking only for rough numbers here....
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