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The circular arc of radius $a$ shown in Figure 3-7 lies in the $x y$ plane and has a constant linear charge density $\lambda$ and center of curvature at the origin. Find $\mathbf{E}$ at an arbitrary point on the $z$ Figure 3-6. The two infinite plane sheets of Exercise 3-9. Figure $3-7 .$ The circular arc of charge of Exercise 3-10. axis. Show that when the curve is a complete circle your answer becomes $$ \mathbf{E}=\frac{\lambda a z \hat{\mathbf{z}}}{2 \epsilon_{0}\left(a^{2}+z^{2}\right)^{3 / 2}} $$

    The circular arc of radius $a$ shown in Figure 3-7 lies in the $x y$ plane and has a constant linear charge density $\lambda$ and center of curvature at the origin. Find $\mathbf{E}$ at an arbitrary point on the $z$
Figure 3-6. The two infinite plane sheets of Exercise 3-9.

Figure $3-7 .$ The circular arc of charge of Exercise 3-10.
axis. Show that when the curve is a complete circle your answer becomes
$$
\mathbf{E}=\frac{\lambda a z \hat{\mathbf{z}}}{2 \epsilon_{0}\left(a^{2}+z^{2}\right)^{3 / 2}}
$$

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Electromagnetic Fields

Electromagnetic Fields

Roald K. Wangsness 1st Edition

Chapter 3, Problem 10

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Step 1

Let's consider a small charge element $dq = \lambda dl$ at an angle $\theta$ from the positive x-axis. The distance from the charge element to the point on the z-axis is $r = \sqrt{a^2 + z^2}$. The electric field produced by this charge element at the point on the Show more…

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