8.1 Capacitors and Capacitance

Capacitors are essential components in electrical systems, storing electric charge and energy. They come in various shapes and sizes, from parallel plates to spheres and cylinders. Understanding their properties is crucial for designing circuits and devices.

Capacitors play a vital role in biology, especially in cell membranes. These biological capacitors influence electrical signaling in neurons and help regulate processes like exocytosis. Studying membrane capacitance provides insights into cellular functions and communication.

Capacitors and Capacitance

Capacitance of various capacitor types

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• Capacitance ($C$) measures a capacitor's ability to store electric charge in response to an applied voltage
  • Defined as the ratio of the charge stored ($Q$) to the applied voltage ($V$): $C = \frac{Q}{V}$
  • Measured in farads (F), where $1 \text{ F} = 1 \text{ C}/\text{V}$ ($1 \text{ [coulomb](https://www.fiveableKeyTerm:coulomb)/volt}$)
• Parallel-plate capacitors consist of two parallel conducting plates separated by a dielectric material (insulator) of thickness $d$
  • Capacitance depends on the plate area ($A$), dielectric constant ($\varepsilon_r$), and plate separation ($d$): $C = \frac{\varepsilon_0 \varepsilon_r A}{d}$, where $\varepsilon_0$ is the permittivity of free space ($8.85 \times 10^{-12} \text{ F/m}$)
  • Example: a parallel-plate capacitor with $A = 100 \text{ cm}^2$, $d = 1 \text{ mm}$, and $\varepsilon_r = 3$ has a capacitance of $26.6 \text{ nF}$
• Spherical capacitors consist of two concentric conducting spherical shells with radii $R_1$ (inner) and $R_2$ (outer)
  • Capacitance depends on the radii and dielectric constant: $C = 4\pi\varepsilon_0 \varepsilon_r \frac{R_1 R_2}{R_2 - R_1}$
  • Example: a spherical capacitor with $R_1 = 5 \text{ cm}$, $R_2 = 10 \text{ cm}$, and $\varepsilon_r = 2$ has a capacitance of $22.1 \text{ pF}$
• Cylindrical capacitors consist of two coaxial conducting cylinders with radii $R_1$ (inner) and $R_2$ (outer) and length $L$
  • Capacitance depends on the radii, length, and dielectric constant: $C = \frac{2\pi\varepsilon_0 \varepsilon_r L}{\ln(R_2/R_1)}$
  • Example: a cylindrical capacitor with $R_1 = 2 \text{ mm}$, $R_2 = 5 \text{ mm}$, $L = 10 \text{ cm}$, and $\varepsilon_r = 4$ has a capacitance of $354 \text{ pF}$

Charge and energy storage in capacitors

• Capacitors store electric charge when a potential difference (voltage) is applied across their conducting plates
  • Positive charge accumulates on one plate, while an equal amount of negative charge accumulates on the other, creating an electric field between the plates
  • The amount of charge stored ($Q$) is proportional to the applied voltage ($V$) and the capacitance ($C$): $Q = CV$
• The electric field between the plates stores electrical potential energy ($U$)
  • Energy stored depends on the capacitance and the applied voltage: $U = \frac{1}{2}CV^2$
  • Example: a $10 \text{ µF}$ capacitor charged to $100 \text{ V}$ stores $50 \text{ mJ}$ of energy
• Capacitors can be connected in series or parallel to achieve desired capacitance values
  • Series connection: the equivalent capacitance ($C_{\text{eq}}$) is the reciprocal of the sum of the reciprocals of the individual capacitances: $\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots$
  • Parallel connection: the equivalent capacitance is the sum of the individual capacitances: $C_{\text{eq}} = C_1 + C_2 + \cdots$
  • Example: two $10 \text{ µF}$ capacitors in series have an equivalent capacitance of $5 \text{ µF}$, while in parallel, their equivalent capacitance is $20 \text{ µF}$
• The charge density on the capacitor plates affects the strength of the electric field between them

Electromagnetism and Capacitors

• The electric field within a capacitor is related to the voltage across its plates and the distance between them
• Maxwell's equations describe the relationship between electric and magnetic fields, including those in capacitors
• The dielectric constant of the material between capacitor plates influences the strength of the electrostatic force between charges

Biological Applications

Capacitance in biological cell membranes

• Cell membranes, composed of a phospholipid bilayer, act as capacitors due to their structure
  • The phospholipid bilayer serves as the dielectric between two conducting surfaces: the extracellular and intracellular fluids
  • Proteins embedded in the membrane contribute to its dielectric properties
• Membrane capacitance ($C_m$) depends on the membrane's surface area ($A$), thickness ($d$), and dielectric constant ($\varepsilon_r$): $C_m = \frac{\varepsilon_0 \varepsilon_r A}{d}$
  • Typical values of membrane capacitance range from $0.5$ to $1.3 \text{ µF/cm}^2$
  • Example: a cell with a surface area of $1000 \text{ µm}^2$ and a membrane capacitance of $1 \text{ µF/cm}^2$ has a total membrane capacitance of $10 \text{ pF}$
• Membrane capacitance plays a crucial role in the propagation of electrical signals in neurons
  • Contributes to the membrane time constant ($\tau_m$), which determines the rate of change of the membrane potential in response to a current: $\tau_m = R_m C_m$, where $R_m$ is the membrane resistance
  • Influences the speed and shape of action potentials, as well as the integration of synaptic inputs
• Changes in membrane capacitance can be used to study exocytosis and endocytosis in cells
  • Fusion of secretory vesicles with the plasma membrane during exocytosis increases the surface area and thus the capacitance
  • Retrieval of membrane through endocytosis decreases the surface area and capacitance
  • Measuring these capacitance changes allows for the quantification of vesicle fusion and retrieval rates
  • Example: capacitance measurements in pancreatic beta cells have revealed that glucose stimulation induces a rapid increase in membrane capacitance, indicating increased insulin secretion through exocytosis