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AP Physics C: E&M
9.1 Electric Potential Energy
AdvancedMCQMathematicalConceptual14.4k
A concentric circular cross-section illustrating the assembly of a sphere. At the center is a solid gray circle of radius \(r\). Enclosing this inner circle is a thin ring of thickness \(dr\) with lighter gray shading and a dashed circular outer boundary. Surrounding both circles is a larger concentric dashed circle of radius \(R\). A horizontal solid arrow points rightward from the center to the edge of the inner solid circle, labeled \(r\). A second solid arrow points from the center upward and rightward at a 45-degree angle to the outer dashed circle, labeled \(R\). A short two-headed radial arrow spans the thickness of the thin ring, labeled \(dr\). No other labels, lines, text, or axes appear.
Cross section showing intermediate radius r, shell thickness dr, and final radius R.
A solid insulating sphere of radius \(R\) has a uniform volume charge density \(\rho\). The sphere is assembled incrementally by an external agent bringing infinitesimal, concentric spherical shells of charge from infinity and depositing them onto the growing sphere until the final radius \(R\) is reached. Taking the electric potential to be zero at infinity, which of the following integral expressions correctly represents the total work \(U\) required to assemble the sphere?

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