= Solution
Use <phasors> with time dependence $e^{i\omega t}$ and <electrical impedance> $Z=\widetilde V/\widetilde I$. The element laws $V=IR$, $I=C\,dV/dt$ and $V=L\,dI/dt$ give
$$
\boxed{Z_R=R,\qquad Z_C=\frac1{i\omega C}=-\frac{i}{\omega C},\qquad Z_L=i\omega L.}
$$
A <complex number> records both <amplitude> and <phase>, turning the linear <differential equations> of a sinusoidally driven <electrical circuit> into algebraic equations. A <resistor> has in-phase <voltage> and <electric current> and dissipates average <power>. For an ideal <capacitor>, the <electric current> leads the <voltage> by $90^\circ$; for an ideal <inductor>, it lags by $90^\circ$. Their imaginary <electrical impedances> express reversible energy storage and return, with zero net average <power> over a period. Using $e^{-i\omega t}$ would conjugate the displayed reactive signs without changing predictions.
In the illustrated <Maxwell bridge>, the left and right vertices are the supply terminals and the upper and lower vertices are detector terminals. Insert a sensitive AC null detector between the latter and drive the former with a sinusoidal source. The upper-left arm is $Z_x=R_x+i\omega L_x$, the upper-right arm is $R_3$, the lower-left arm is $R_2$, and the lower-right arm is the parallel combination
$$
Z_p=\left(\frac1{R_1}+i\omega C\right)^{-1}.
$$
At a null no detector <electric current> flows, so the two voltage dividers give
$$
\frac{R_3}{Z_x+R_3}=\frac{Z_p}{R_2+Z_p},\qquad
Z_xZ_p=R_2R_3.
$$
Multiplying by the parallel-arm <electrical admittance> yields
$$
R_x+i\omega L_x=R_2R_3\left(\frac1{R_1}+i\omega C\right).
$$
Equating real and imaginary parts therefore gives
$$
\boxed{R_x=\frac{R_2R_3}{R_1},\qquad L_x=CR_2R_3.}
$$
Two independent adjustments are needed to annul the in-phase and quadrature detector signals. For example, change $R_2$ to set $L_x$ and $R_1$ to set $R_x$, iterating until the detector reads zero. Alternatively, with fixed $R_2,R_3$, independently vary $R_1$ and $C$.
The <Maxwell bridge> compares an unknown lossy <inductor> with a stable standard <capacitor>, avoiding the need for an accurately known reference <inductor>. The ideal balance equations contain no <frequency>, and a null measurement need not rely on absolute detector calibration. In practice choose a suitable <frequency> below the coil's self-resonance, where the series $R_x,L_x$ model and the standard <capacitor> remain accurate.
If the initial setting is far from balance, the bridge as drawn needs adjustable components and a detector, not merely a more precise readout. Fit resistance boxes or switched ranges for $R_1$ and a ratio arm, and a selectable standard $C$. Choose ranges around
$$
R_2R_3=\frac{L_x}{C},\qquad R_1=\frac{L_x}{CR_x}.
$$
Then approach the null with coarse adjustments followed by finer ones. A large required $R_1$ can be brought into range by increasing $C$ and decreasing $R_2R_3$ proportionally. Start with reduced excitation or reduced detector sensitivity to avoid overload, then increase sensitivity near the null. Changing <frequency> alone cannot repair an impossible component range, since the ideal balance conditions are frequency-independent. Stray <capacitance> and lead <electrical resistance> set practical accuracy limits and may require shielding and short connections.
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