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## Homework Statement

The linear charge density for a ring of radius R which lies in the xy-plane (center at the origin) is given by:[tex]\eta(\phi) = \left\{

\begin{array}{c l}

+\lambda & \mbox{if } 0<\phi<\frac{\pi}{2} \\

- \lambda & \mbox{if } \frac{\pi}{2}<\phi<\pi \\

+\lambda & \mbox{if } \pi < \phi < \frac{3\pi}{2} \\

-\lambda & \mbox{if } \frac{3\pi}{2} < \phi < 2\pi

\end{array}

\right.[/tex]

where [tex]\phi[/tex] is the usual azimuthal angle and [tex]\lambda > 0[/tex] is a constant of appropriate dimensions. Discuss the electrostatics of this system, in particular:

a) Show a figure with the electric field lines in the xy-plane;

b) Calculate the electric field along the x-axis for both [tex]|x| < R[/tex] and [tex]|x| > R[/tex]

c) an expression for the far-field; i.e., when [tex]|x| >> R[/tex]

## Homework Equations

Electric field: [tex]E = \frac{kQ}{r^2}[/tex]

Electric field along the axis of a ring of radius R with uniform charge: [tex] E = \frac{kQx}{(x^2+R^2)^{\frac{3}{2}}}[/tex], where x is the distance along the axis from the center of the ring.

## The Attempt at a Solution

To tell you the truth, I am not really sure where to start. I drew a picture of the quadrupole, but what is confusing me is that the example we did in class involved the electric field along the axis of the ring. Here, the ring lies in the xy-plane, which is confusing me. The axis in this case would be the z-axis, not the x-axis.

Nevertheless, I first tried inside the ring along the x-axis. I chose an arbitrary point (not on the origin). Then I wanted to find the distances from each of the segments of the ring. But, that become difficult because I couldn't draw any triangles (the boundary is a circle) like we did with the example in class. I tried the same on the outside of the circle, but I ran into the same problem. That is, I cannot figure out how to draw the shapes needed since the ring lies in the xy-plane.

What I want to do is first find the equation for the electric field and then use that to draw my electric field lines. If you could provide any help to get me started on finding the equations, that would be great. Thank you.