Problem Set 8 (Given 12/18/11)

Level 1

  1. Two small \(1 kg\) spheres sit isolated in space 1 meter apart. Each sphere has charge \(+Q\). For what value of \(Q\) is the net force on each of the masses zero?
  2. For the above configuration, draw the electric field lines and the equipotential lines.
  3. Four point charges are fixed on the corners of a square with side length \(d\). If all four have charge \(-q\), what is the electric potential in the center of the square? What if two have charge \(+q\) and two have charge \(-q\)?
  4. How much energy is required to create the configuration in #3 in each of the two cases?

Level 2

  1. You hang a block of charge \(q\) and mass \(m\) from a string in the presence of a uniform electric field parallel to the ground. You notice that the block stays at rest when the string is displaced from the vertical by an angle \(\theta\). What is \(E\), the magnitude of the uniform electric field?
  2. A rod of uniform charge density is situated on the $x$-axis from \(x=0\) to \(x=L\). The rod has total charge \(+Q\). What is the electric field at a point along the $x$-axis for \(x > L\)? What happens as \(x\) becomes much larger than \(L\)?
  3. Suppose we have a uniformly charged ring of radius \(R\) and charge \(Q\). Calculate the electric field and the electric potential directly in the center of the ring. Now suppose the bottom half of the ring were removed. Would the magnitude of the electric field at the same point increase or decrease? How about the electric potential?
  4. Two point charges lie on the $y$-axis. A charge of \(+q\) is located at \(y=+1m\), and a charge of \(-q\) is located at \(y=-1m\). What is the electric field at an arbitrary point \((x,y)\) in this space?

Level 3

  1. Suppose we have a disk of radius \(R\) in the $xy$-plane centered at the origin with a uniform charge density and total charge \(Q\). By treating the disk as a series of concentric rings, find the electric field at an arbitrary point along the $z$-axis.

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