Problem 6: Problem 23.80 in Young & Freedman An annulus with an inner radius of a and an outer radius of b has charge density ? and lies in the xy-plane with its center at the origin (see the figure). (a) Using the convention that the potential vanishes at infinity, determine the potential at all points on the z-axis. (b) Determine the electric field at all points on the z-axis by differentiating the potential. (c) Show that in the limit a ? 0, b ? ?, the electric field reproduces the result for an infinite plane sheet of charge. (d) If a = 5.00 cm, b = 10.0 cm and the total charge on the annulus is 1.00 ?C, what is the potential at the origin? (e) If a particle with mass 1.00 g (much less than the mass of the annulus) and charge 1.00 ?C is placed at the origin and given the slightest nudge, it will be projected along the z-axis. In this case, what will be its ultimate speed?
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The total potential at P due to the entire annulus is obtained by integrating this expression over the entire annulus. The result is V = kQ / sqrt(z^2 + b^2) - kQ / sqrt(z^2 + a^2), where Q is the total charge on the annulus. Show more…
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A thin disk with a circular hole at its center, called an $annulus$, has inner radius $R_1$ and outer radius $R_2 \space (\textbf{Fig. P21.91}$). The disk has a uniform positive surface charge density $\sigma$ on its surface. (a) Determine the total electric charge on the annulus. (b) The annulus lies in the $yz$-plane, with its center at the origin. For an arbitrary point on the $x$-axis (the axis of the annulus), find the magnitude and direction of the electric field $\overrightarrow{E}$. Consider points both above and below the annulus. (c) Show that at points on the $x$-axis that are sufficiently close to the origin, the magnitude of the electric field is approximately proportional to the distance between the center of the annulus and the point. How close is "sufficiently close"? (d) A point particle with mass $m$ and negative charge $-q$ is free to move along the $x$-$axis$ (but cannot move off the axis). The particle is originally placed at rest at $x = 0.01 \space R_1$ and released. Find the frequency of oscillation of the particle. ($Hint$: Review Section 14.2. The annulus is held stationary.)
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