Homework for Static Fields

  1. (BiotSavartSquare) Calculate the magnetic field above a square sheet of current using the Biot-Savart Law. Then take the limit as the square becomes infinite in size.

    Consider a point a distance $z$ above the center of an infinitesimally thin, square sheet of current. The current is parallel to one of the square sides. (Obviously, since the current cannot just begin and end in the middle of nowhere, this current is just the building block for some larger current.)

    1. Use the Biot-Savart Law to find the magnetic field at the point $z$. You may use any symmetry arguments you like, but do not use Ampere's Law.

      Note: if you choose to use Mathematica or Maple to evaluate the integral, it may take you into complex number land, even though the integral is clearly real. To address this issue, you should be explicit about what assumptions you want the program to make (“Assume” in Maple and “Assumptions” in Mathematica)

    2. Consider your previous answer in the limit that the square becomes infinitely large.

    3. Discuss your answer in the light of the magnetic field above an infinite sheet of current as found using Ampere's Law.

  2. (BFiniteLine)

    1. Find the magnetic field for a finite segment of straight wire, carrying a uniform current $I$. Put the wire on the $z$ axis, from $z_1$ to $z_2$.

    2. Show that your answer to part (a) is the curl of the magnetic vector potential.

  3. (BiotSavartCoil)

    Two charged rings of radius $R$ spin in opposite directions, each with total current $I$. They are placed a distance $2L$ apart and oriented as shown below.

    \medskip \centerline{\includegraphics[scale=0.65]{\TOP Figures/vfbiotsavartcoils}} \medskip

    1. What is the magnetic field on the $z$-axis due to Loop 1?

    2. What is the magnetic field on the $z$-axis due to Loop 2?

    3. What is the leading non-zero term for the total magnetic field on the $z$-axis near the midpoint between the coils ($z«R$)?

  4. (BiotSavartChallenge)

    In class, we found that the magnetic vector potential created by a rotating ring of charge (total charge $Q$, radius $R$, rotating with period $T$) everywhere in space is\\

    \begin{equation*} \vec{A}(\vec{r}) =\frac{\mu_0}{4 \pi}\frac{Q\,R}{T}\,\hat{\phi}\int_0^{2 \pi} \dfrac{\cos\phi'\,d\phi'}{\sqrt{r^2+R^2-2 r R cos\phi'+z^2}} \end{equation*}\\

    Show that $\vec{B}=\vec{\nabla}\times\vec{A}$ for this spinning ring (\emph{i.e.} show that that the curl of this expression for $\vec{A}$ equals the expression for $\vec{B}$ in Problem 1)


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