Electric Field and Charge

  • divergence charge density Maxwell's equations electric field
  • Consider the electric field \begin{equation} \vec E(r,\theta,\phi) = \begin{cases} 0&\textrm{for } r<a\\ \frac{1}{4\pi\epsilon_0} \,\frac{Q}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ 0 & \textrm{for } r>b \\ \end{cases} \end{equation}
    1. (4pts) Use step and/or delta functions to write this electric field as a single expression valid everywhere in space.
    2. (4pts) Find a formula for the charge density that creates this electric field.
    3. (2pts) Interpret your formula for the charge density, i.e. explain briefly in words where the charge is.