Problem Statement
Solve the capacitor/capacitance problem: Solve the capacitor/capacitance problem: Show that a pure inductor and a pure capacitor consume zero average power from an AC source. Pure inductor ($R=0$, $X_L = \omega L$): current lags voltage by $90°$, i.e., $\psi = -\pi/2$. $$\langle P_L\rangle = V_{\rm rms}I_{\rm rms}\cos(-\pi/2) = 0$$ Pure ca
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
Capacitors store electric charge on conducting plates separated by an insulator (dielectric). The capacitance $C = Q/V$ depends on geometry and dielectric constant. Energy stored is $U = Q^2/(2C) = CV^2/2 = QV/2$. Series and parallel combinations follow rules opposite to resistors.
- $C = Q/V$ — definition of capacitance
- $U = \frac{1}{2}CV^2 = \frac{Q^2}{2C}$ — energy stored
- $C_{\text{parallel}} = C_1 + C_2$ — parallel combination
- $1/C_{\text{series}} = 1/C_1 + 1/C_2$ — series combination
- $C = \varepsilon_0\varepsilon_r A/d$ — parallel plate capacitor
Step-by-Step Solution
Step 1 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 2 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Worked Calculation
$$\langle P_L\rangle = V_{\rm rms}I_{\rm rms}\cos(-\pi/2) = 0$$
$$C = \frac{\kappa\varepsilon_0 A}{d} = \frac{2\times8.85\times10^{-12}\times0.02}{10^{-3}} = \frac{3.54\times10^{-13}}{10^{-3}} = 3.54\times10^{-10}\,\text{F} = 354\,\text{pF}$$
$$\boxed{C = \dfrac{\kappa\varepsilon_0 A}{d}}$$
Answer
$$\boxed{C = \dfrac{\kappa\varepsilon_0 A}{d}}$$
Physical Interpretation
Capacitors store energy in the electric field between their plates. Doubling the voltage quadruples the stored energy — an important design constraint for high-voltage applications. Charge sharing between capacitors is a lossless process only in the ideal case; real circuits dissipate energy in connecting resistance.
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