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AP Physics C: E&M
12.3 Magnetic Fields of Current-Carrying Wires and the Biot-Savart Law
AdvancedMCQMathematicalProportional AnalysisConceptual18.1k
A flat circular wire loop is drawn as an ellipse in perspective, positioned horizontally in the lower half of the frame. A curved arrow on the loop indicates counterclockwise current labeled I when viewed from above. A straight line segment extends from the center of the loop toward the upper-right edge of the loop, labeled R. A vertical dashed line passes through the geometric center of the loop, extending upward to represent the central axis, labeled z at the top. A distinct dot marks an axial point on this vertical line above the center; a double-headed vertical dimension line between the loop center and the dot is labeled z. A solid straight arrow begins at the dot and points straight upward along the dashed vertical line, labeled \vec{B}. No other labels, lines, text, or axes appear.
Circular current loop of radius \(R\) and axial magnetic field \(\vec{B}\) at distance \(z\).
Students measure the magnetic field magnitude \(B\) along the central axis of a circular wire loop of radius \(R\) carrying a steady current \(I\), as a function of the axial distance \(z\) from the center of the loop.

According to the Biot-Savart law, the on-axis magnetic field is given by
\[ B(z) = \dfrac{\mu_0 I R^2}{2(R^2 + z^2)^{3/2}} \]

The students wish to construct a single linear graph using all their data points (\(z \ge 0\)) to confirm this functional relationship, which approaches the inverse-cube dipole dependence \(B \propto z^{-3}\) for \(z \gg R\). Which of the following pairs of quantities should be plotted on the vertical and horizontal axes to produce a linear graph, and what relationship correctly yields the radius \(R\) from the vertical intercept \(b\) and slope \(m\) of the best-fit line?

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