NBPhO 2026

A circuit has four nodes $A$, $B$, $E$, $D$. An AC source of amplitude $V$ is applied between $A$ and $E$. A resistor $R_1$ connects $A$ to $B$ and a capacitor $C$ connects $B$ to $E$; a resistor $R_2$ connects $A$ to $D$ and an inductor $L$ connects $D$ to $E$. A potentiometer with a sliding contact $P$ is connected between $A$ and $E$. The voltage between $P$ and $B$ feeds the $x$-input of an oscilloscope, and the voltage between $P$ and $D$ feeds the $y$-input; both input channels have the same gain. The oscilloscope plots the voltage $V_x$ from the $x$-input and the voltage $V_y$ from the $y$-input in the $V_x$–$V_y$-plane. The position of the sliding contact $P$ is adjusted until the figure on the screen degenerates into a line segment. It appears that in that case, the line segment makes an angle $\alpha$ with the $x$-axis, and the sliding contact $P$ divides the potentiometer’s length in the ratio $1 : 2$ from left to right in the figure. Find the amplitude of the voltage between $B$ and $P$.