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5: Capacitors

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    5444

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    • 5.1: Introduction
      A capacitor consists of two metal plates separated by a nonconducting medium (known as the dielectric medium or simply the dielectric) or by a vacuum.
    • 5.2: Plane Parallel Capacitor
      This page covers capacitor properties, focusing on key parameters: plate area (A), separation (d), and permittivity (ε). It explains how the electric field (E) is derived from the EMF (V) as E = V/d, and describes the relationship between charge (Q) and capacitance (C) formulated as C = εA/d. The importance of these variables on capacitance is highlighted, and the page discusses permittivity units and poses a question regarding the effects of plate separation on electric field uniformity.
    • 5.3: Coaxial Cylindrical Capacitor
      This page provides an overview of the capacitance per unit length for coaxial cylinders, characterized by inner and outer radii \(a\) and \(b\), and the permittivity \(\epsilon\) between them. It derives the potential difference \(V\) as \(\frac{\lambda}{2\pi \epsilon} \ln (b/a)\) when charge distributions are applied. The capacitance formula is given as \(C' = \frac{2\pi \epsilon}{\ln (b/a)}\), emphasizing its importance in coaxial cable design.
    • 5.4: Concentric Spherical Capacitor
      This page covers the capacitance of a spherical capacitor formed by two concentric spheres, detailing the derivation of potential difference and capacitance formula \(C\). It discusses the scenario where the outer sphere's radius approaches infinity to determine the capacitance of an isolated sphere. An exercise is included to calculate Earth's capacitance, yielding 709 μF and encouraging consideration of the practical implications of capacitance in real-world applications.
    • 5.5: Capacitors in Parallel
      For capacitors in parallel, the potential difference is the same across each, and the total charge is the sum of the charges on the individual capacitor.
    • 5.6: Capacitors in Series
      The potential difference across the system of capacitors in series is the sum of the potential differences across the individual capacitances.
    • 5.7: Delta-Star Transform
      We can make a delta-star transform with capacitors.
    • 5.8: Kirchhoff’s Rules
      This page explains the application of Kirchhoff’s rules to capacitor circuits with a 24 V battery. It details the steps to calculate charge on each capacitor by forming equations from Kirchhoff's laws and presents relationships between various charges (Q1-Q5). The page concludes with the calculated values for the charges on the capacitors, demonstrating the practical use of Kirchhoff's principles in analyzing capacitor circuits.
    • 5.9: Problem for a Rainy Day
      This page explores replacing cube resistors with capacitors of equal capacitance \(c\) and calculates the total capacitance between opposite corners. It uses an assumed net charge of \(6Q\) to analyze charge distribution in the capacitors, ultimately concluding that the effective capacitance of the cube is 1.2\(c\).
    • 5.10: Energy Stored in a Capacitor
      This page explains how energy is stored in a capacitor, detailing the relationship between capacitance \(C\), charge \(+q\) and \(-q\), and the potential difference \(q/C\). It describes the work needed to move a small charge \(+\delta q\), culminating in the total work to transfer charge \(Q\), represented by the formula \(U = \frac{1}{2} CV^2\). The page also validates the dimensions of these expressions, reinforcing the connection between charge, capacitance, and energy storage.
    • 5.11: Energy Stored in an Electric Field
      This page presents the fundamentals of capacitance and energy storage in capacitors, highlighting the uniform electric field in closely spaced plates. It defines capacitance as \(C = \epsilon A/d\) and describes energy storage in dielectric materials and vacuum with specific equations. The page also addresses the complexities of energy density in anisotropic media, where permittivity is tensor-based, requiring different formulations involving the vectors \(\textbf{D}\) and \(\textbf{E}\).
    • 5.12: Force Between the Plates of a Plane Parallel Plate Capacitor
      This page explores the dynamics of a capacitor with charged plates, detailing the force and energy relationships when the plates are separated. It discusses the work done in separating the plates, correlating it to the energy stored in the capacitor, and includes an experimental setup with a spring-connected upper plate. The analysis reveals two equilibrium positions at voltages below a specific threshold, highlighting stability and instability in the system's plate positions.
    • 5.13: Sharing a Charge Between Two Capacitors
      This page examines the behavior of two interconnected capacitors, \(C_1\) and \(C_2\), after \(C_1\) is charged. It discusses how charge redistributes between them and calculates the energy stored in each capacitor.
    • 5.14: Mixed Dielectrics
      This page explores the behavior of electric fields in capacitors with various dielectric materials, highlighting the continuity of the electric displacement field \(D\) across boundaries while the electric field \(E\) varies based on permittivity. In series configurations, \(D\) stays constant but \(E\) differs between materials, whereas in parallel setups, \(E\) is uniform while charge densities change according to permittivities.
    • 5.15: Changing the Distance Between the Plates of a Capacitor
      This page explains the effects of increasing the plate separation in a capacitor on the electric field \(E\) and displacement field \(D\). For isolated plates, both fields remain constant while potential difference rises, leading to decreased capacitance and more stored energy. For connected plates, \(E\) declines with increased separation, resulting in reduced charge as the battery compensates.
    • 5.16: Inserting a Dielectric into a Capacitor
      This page explains the impact of inserting a dielectric between parallel plates on the electric field, charge, and potential. With isolated plates, the electric field diminishes due to polarization, reducing stored energy. When connected to a battery, the potential remains constant, increasing the D-field, charge density, and capacitance, which raises stored energy. The process of removing the dielectric or charging the battery involves work, affecting current flow in the circuit.
    • 5.17: Polarization and Susceptibility
      This page explains how insulating materials experience polarization in an electric field, detailing the alignment of molecules and the creation of dipole moments. It introduces the equation \(D = \epsilon_0 E + P\) relating electric displacement, electric field, and polarization. Polarization, defined as the dipole moment per unit volume, is linked to electric susceptibility \(\chi_e\), which is related to the dielectric constant \(\epsilon_r\) through the formula \(\chi_e = \epsilon_r - 1\).
    • 5.18: Discharging a Capacitor Through a Resistor
      This page discusses the significance of accurately identifying signs in capacitor charge dynamics, introducing the rate of change of charge during discharge. It establishes the connection between charge, current, and potential difference across capacitors and resistors.
    • 5.19: Charging a Capacitor Through a Resistor
      This page explains capacitor charging behavior in a circuit, highlighting the relationship between charge, current, and potential difference with specific timeframes for charge realization. It notes the initial current spike followed by exponential decay influenced by circuit inductance. Energy analysis shows equal energy loss by the battery between the resistor and capacitor. Finally, it describes the operation of a neon lamp, which discharges at a certain threshold potential.
    • 5.20: Real Capacitors
      Real capacitors can vary from huge metal plates suspended in oil to the tiny cylindrical components seen inside a radio. A great deal of information about them is available on the Web and from manufacturers’ catalogs, and I only make the briefest remarks here.
    • 5.21: More on E, D, P, etc
      This page covers electric fields and charge distributions in capacitors, focusing on dielectrics and their effects on polarization. It outlines key concepts such as the relationship between the electric field and the displacement field, the differences between permanent and induced dipoles, and the influence of temperature on dipole alignment via the Langevin function.
    • 5.22: Dielectric material in an alternating electric field.
      This page explores dielectric polarization in electric fields, highlighting the significance of relaxation time, which varies across materials. It explains the phase lag between the displacement field \(D\) and electric field \(E\) during oscillating fields, introducing complex permittivity to address this difference.

    Thumbnail: Capacitors connected in series. The magnitude of the charge on each plate is Q. (CC BY-SA 3.0; OpenSTAX).

    This page titled 5: Capacitors is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.