Problem Set 9 (Given 01/08/12)

Level I

  1. Find the three partial derivatives \(\frac{\partial}{\partial x}\), \(\frac{\partial}{\partial y}\), and \(\frac{\partial}{\partial z}\) of \(f(x, y, z) = xyz + x^2 y + y^2 z + z^2 x - xy - yz - xz\).
  2. Find the path integral along the unit circle in the \(xy\) plane of \(F(x, y, z) = (x + 3) {\hat x} + (y + 2) {\hat y} + (z - 1) {\hat z}\).
  3. Convince yourself that work is the path-integral of force. Formalize what this means.

Level II

  1. Find the electric field inside and outside a shell of charge. This is often referred to as the "conducting sphere" case — "conducting" objects only have charge on their boundary.
  2. Find the electric field inside a plane of finite thickness (not infinitely thin).
  3. We discussed in class that when charge is uniformly distributed throughout a sphere, the electric field strength increases linearly as distance from the center increases (while you are within the sphere). Could you, and if yes how could you, distribute charge within a sphere so that the electric field strength inside the sphere increases quadratically with radius (\(E \sim r^2\)).

Level III

  1. Let the charge density in all space be \(\rho = \rho_0 / r^2 (r + 1)^2\). Calculate the electric field at an arbitrary point in space.
  2. Calculate the electric field due to a line of charge by integrating over Coulomb's law. Compare to using Gauss's law.

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