= Solution
For <electric potential difference> and <electric current> varying as <sinusoids>, choose the <phasor> convention
$$
V(t)=\operatorname{Re}(\widetilde V e^{i\omega t}),\qquad I(t)=\operatorname{Re}(\widetilde I e^{i\omega t}).
$$
The <electrical impedance> is
$$
\boxed{Z(\omega)=\frac{\widetilde V}{\widetilde I},\qquad [Z]=\Omega.}
$$
Its <modulus> gives the voltage-to-current <amplitude> ratio, and its <complex argument> gives the voltage-minus-current <phase difference>. For ideal components,
$$
\boxed{Z_R=R,\qquad Z_C=\frac1{i\omega C}=-\frac{i}{\omega C},\qquad Z_L=i\omega L.}
$$
These follow from <Ohm's law> $V=RI$, the <capacitor> law $I=C\,dV/dt$, and the <inductor> law $V=L\,dI/dt$. In <phasor> notation a <time derivative> becomes multiplication by $i\omega$, turning the <linear ordinary differential equations> of the <electrical circuit> into algebraic equations. <Electrical impedances> add in a <series circuit>; their reciprocals, the <electrical admittances>, add in a <parallel circuit>. This computes both <amplitudes> and <phase differences> in the steady response to <sinusoids>.
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