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AP Physics C: E&M
8.6 Gauss’s Law
8.4 Electric Fields of Charge Distributions
AdvancedMCQMathematical16.1k
Cross-section of a thick spherical shell, drawn as three concentric circles sharing one center, marked with a small black dot. The inner circle has radius R; the outer circle has radius exactly twice the inner circle's; a dashed middle circle lies between them at radius r. The annular region between the inner and outer circles is shaded light gray to represent the charged material; the region inside the inner circle is empty white. Draw exactly three radius arrows, all starting at the center dot: a solid arrow toward the upper-right (45 degrees) ending with its arrowhead touching the inner circle, labeled R at the arrow's midpoint; a solid arrow toward the upper-left (135 degrees) ending with its arrowhead touching the outer circle, labeled 2R at its midpoint; and a solid arrow pointing straight down ending with its arrowhead touching the dashed circle, labeled r at its midpoint. Exactly one arrow and one label per radius value — no duplicate radius lines, no tick marks, no extra labels, and no text or formulas anywhere else in the figure.
Cross-section of a non-uniform spherical shell and a Gaussian surface at radius r.
A thick, non-conducting spherical shell has an inner radius \(R\) and an outer radius \(2R\). The shell has a non-uniform volume charge density \(\rho(r) = \rho_0 \left(\dfrac{r}{R}\right)\) for \(R < r < 2R\), where \(\rho_0\) is a positive constant and \(r\) is the distance from the center of the shell. The region \(r < R\) is empty. Which of the following expressions represents the magnitude of the electric field \(E\) at a distance \(r\) from the center, where \(R < r < 2R\)?

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