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AP Physics C: E&M
9.2 Electric Potential
9.1 Electric Potential Energy
AdvancedMCQMathematical13.1k
A horizontal axis labeled with an italic letter x extends to the right with an arrowhead. A solid black dot on the axis at the far left is labeled with a plus sign followed by Q, positioned above a vertical tick mark labeled 0. Further to the right along the horizontal axis, a filled circle is labeled with a minus sign followed by q, located above a tick mark labeled r. A double-headed horizontal arrow below the axis extends from the tick mark at 0 to the tick mark at r, with the label r centered beneath it. Further to the right of the minus charge, another filled circle is labeled with a plus sign followed by q, located above a tick mark labeled r + s. A smaller double-headed horizontal arrow spans the interval between the two charges with the label s centered above it. No other labels, lines, text, or axes appear.
Point charge \(+Q\) and physical dipole aligned along the \(x\)-axis.
A point charge \(+Q\) is fixed at the origin. A physical electric dipole aligned along the \(x\)-axis consists of charge \(-q\) at \(x = r\) and charge \(+q\) at \(x = r + s\), where \(s\) is a fixed positive separation distance and \(r > 0\). The electrostatic potential energy of the dipole due to the charge \(+Q\) is given by \(U(r) = kQq\left(\dfrac{1}{r + s} - \dfrac{1}{r}\right)\). In the limit where the distance to the dipole is much greater than its size (\(r \gg s\)), which of the following correctly gives the leading-order expression for \(U(r)\) including its first non-vanishing correction term in terms of the dipole moment \(p = qs\), and correctly compares \(U(r)\) to the ideal point-dipole potential energy \(U_{\text{ideal}} = -\dfrac{kQp}{r^2}\)?

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