Capacitance Equations

Why This Matters

Capacitance equations aren't just formulas to memorize—they reveal the fundamental relationship between geometry, materials, and energy storage in electric systems. You're being tested on your ability to recognize how plate area, separation distance, and dielectric materials each influence a capacitor's behavior, and why different configurations (series vs. parallel) produce opposite effects on total capacitance.

These equations connect directly to broader electromagnetism principles: Gauss's law explains the electric field behavior, energy density concepts tie into field theory, and dielectric polarization bridges microscopic material properties with macroscopic circuit behavior. Don't just memorize the formulas—know what physical principle each equation represents and when to apply it.

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Fundamental Definitions

These equations establish what capacitance is and how we quantify energy storage. Capacitance measures a system's ability to store separated charge for a given potential difference.

Capacitance Definition

• $C = Q/V$—capacitance equals stored charge divided by voltage across the capacitor
• Farads (F) are the SI unit, where $1 \text{ F} = 1 \text{ C/V}$; most practical capacitors are measured in microfarads or picofarads
• Higher capacitance means more charge stored at the same voltage—this is the core concept for all capacitor problems

Energy Stored in a Capacitor

• $U = \frac{1}{2}CV^2$—energy scales with capacitance and the square of voltage
• Equivalent forms: $U = \frac{1}{2}QV = \frac{Q^2}{2C}$—use whichever matches your known quantities
• Measured in joules (J); the quadratic voltage dependence explains why high-voltage capacitors store dramatically more energy

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Geometry-Dependent Capacitance

Different capacitor shapes require different formulas, but they all share a common principle: capacitance increases when you maximize the area where charge accumulates and minimize the separation between opposite charges.

Parallel Plate Capacitor

• $C = \frac{\varepsilon A}{d}$—capacitance is directly proportional to plate area and inversely proportional to separation
• $\varepsilon$ is the permittivity of the material between plates; for vacuum, use $\varepsilon_0 = 8.85 \times 10^{-12} \text{ F/m}$
• Doubling area doubles capacitance; halving distance doubles capacitance—know both relationships for quick problem-solving

Cylindrical Capacitor

• $C = \frac{2\pi\varepsilon L}{\ln(b/a)}$—depends on length $L$ and the natural log of the radius ratio
• $a$ is inner radius, $b$ is outer radius; the logarithm appears because the electric field varies with radial distance
• Longer cylinders increase capacitance linearly; used in coaxial cables and cylindrical sensors

Spherical Capacitor

• $C = 4\pi\varepsilon \left(\frac{ab}{b-a}\right)$—alternatively written as $C = \frac{4\pi\varepsilon}{\frac{1}{a} - \frac{1}{b}}$
• $a$ is inner sphere radius, $b$ is outer sphere radius; field geometry is radial
• Isolated sphere capacitance: when $b \to \infty$, this simplifies to $C = 4\pi\varepsilon_0 a$—useful for modeling charged spherical conductors

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Capacitor Combinations

When capacitors connect in circuits, total capacitance follows rules that are opposite to resistor combination rules. The key is understanding what stays constant: voltage (parallel) or charge (series).

Capacitors in Parallel

• $C_{\text{total}} = C_1 + C_2 + C_3 + \ldots$—total capacitance is the direct sum of individual values
• Voltage is identical across all capacitors; charges add together
• Increases total capacitance—parallel combinations are used when you need more charge storage at a fixed voltage

Capacitors in Series

• $\frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \ldots$—reciprocals add, so total is always less than the smallest individual capacitor
• Charge is identical on each capacitor; voltages add together
• Decreases total capacitance—series combinations are used to handle higher total voltage or reduce effective capacitance

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Dielectric Effects

Inserting an insulating material between capacitor plates changes everything. Dielectrics polarize in the external field, reducing the net field and allowing more charge to accumulate at the same voltage.

Dielectric Constant Relationship

• $C = \kappa C_0$—capacitance with dielectric equals the dielectric constant times the vacuum capacitance
• $\kappa$ (kappa) is always ≥ 1; air is approximately 1, water is ~80, ceramics range from 10–10,000
• Physical mechanism: molecular dipoles align and partially cancel the applied field, requiring more surface charge to maintain the same voltage

Parallel Plates with Dielectric

• $C = \frac{\kappa \varepsilon_0 A}{d}$—the standard parallel plate formula multiplied by the dielectric constant
• $\varepsilon_0$ is the permittivity of free space; $\kappa \varepsilon_0 = \varepsilon$ gives the material's total permittivity
• Practical applications: dielectrics increase capacitance without changing physical dimensions, enabling compact high-capacitance components

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Field Theory Foundation

Gauss's law provides the theoretical basis for deriving all capacitance formulas. It connects the macroscopic charge on plates to the microscopic electric field between them.

Gauss's Law for Capacitors

• $\Phi_E = \frac{Q}{\varepsilon_0}$—electric flux through a closed surface equals enclosed charge divided by permittivity
• Derivation tool: by choosing Gaussian surfaces that match capacitor symmetry, you can find the electric field and then integrate to get capacitance
• Connects to all geometry formulas—parallel plate, cylindrical, and spherical capacitance equations all derive from applying Gauss's law with appropriate symmetry

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Quick Reference Table

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Self-Check Questions

1. If you double the plate separation in a parallel plate capacitor while keeping everything else constant, what happens to capacitance and stored energy (assuming charge is held constant)?

2. Which capacitor geometry—parallel plate, cylindrical, or spherical—would you use to model a coaxial cable, and what variables determine its capacitance?

3. Compare and contrast capacitors in series vs. parallel: which arrangement increases total capacitance, and what quantity (charge or voltage) remains the same across all capacitors in each case?

4. A dielectric with $\kappa = 4$ is inserted into a capacitor that remains connected to a battery. By what factor do the capacitance, charge, and stored energy change?

5. Using Gauss's law, explain why the electric field between parallel plates is uniform while the field in a cylindrical capacitor varies with radial distance.