Homework for Static Fields

  1. (Capacitor) Examine the electric field of a parallel plane capacitor through superposition.

    We know that the electric field everywhere in space due to an infinite plane of charge with charge density located in the $xy$-plane at $z=0$ is \begin{equation*} \EE(z) = \begin{cases}\displaystyle +{\sigma\over2\epsilon_0}\>\zhat & z>0 \cr \noalign{\smallskip}\displaystyle -{\sigma\over2\epsilon_0}\>\zhat & z<0 \end{cases} \end{equation*}

    (Mentally check that this is true for both positive and negative values of $\sigma$.)

    1. Sketch the $z$-component of the electric field as a function of $z$.

    2. Draw a similar picture, and write a function that expresses the electric field everywhere in space, for an infinite conducting slab in the $xy$-plane, of thickness $d$ in the $z$-direction, that has a charge density $+|\sigma|$ on each surface.

    3. Repeat for a charge density $-|\sigma|$ on each surface.

    4. Now imagine two {\bf conductors}, one each of the two types described above, separated by a distance $L$. Use the principle of superposition to find the electric field everywhere. Discuss whether your answer is reasonable. Does it agree with the known fact that the electric field inside a conductor is zero? Has superposition been correctly applied? Is Gauss' Law correct? Try to resolve any inconsistencies.

  2. (ConductorsGEM235) A long problem about the charge density, potential, and electric field due to a conducting sphere surrounded by a conducting shell, from Griffiths E&M book.

    A metal sphere of radius $R$, carrying charge $q$ is surrounded by a thick concentric metal shell (inner radius $a$, outer radius $b$, as shown below). The shell carries no net charge.

    \medskip \centerline{\includegraphics[scale=1]{\TOP Figures/vfconductor}} \medskip

    1. Find the surface charge density $\sigma$ at $R$, at $a$, and at $b$.

    2. Find $E_r$, the radial component of the electric field and plot it as a function of $r$. Are the discontinuities in the electric field related to the charge density in the way you expect from previous problems?

    3. Find the potential at the center of the sphere, using infinity as the reference point.

    4. Now the outer surface is touched to a grounding wire, which lowers its potential to zero (the same as infinity). How do your answers to a), b), and c) change?


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