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 . These biological capacitors influence electrical signaling in neurons and help regulate processes like . Studying provides insights into cellular functions and communication.
Capacitors and Capacitance
Capacitance of various capacitor types
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19.5 Capacitors and Dielectrics – College Physics: OpenStax View original
(C) measures a 's ability to store electric charge in response to an applied
Defined as the ratio of the charge stored (Q) to the applied voltage (V): C=VQ
Measured in farads (F), where 1 F=1 C/V (1 [coulomb](https://www.fiveableKeyTerm:coulomb)/volt)
Parallel-plate capacitors consist of two parallel conducting plates separated by a material (insulator) of thickness d
depends on the plate area (A), constant (εr), and plate separation (d): C=dε0εrA, where ε0 is the of free space (8.85×10−12 F/m)
Example: a with A=100 cm2, d=1 mm, and εr=3 has a capacitance of 26.6 nF
Spherical capacitors consist of two conducting spherical shells with radii R1 (inner) and R2 (outer)
Capacitance depends on the radii and dielectric constant: C=4πε0εrR2−R1R1R2
Example: a with R1=5 cm, R2=10 cm, and εr=2 has a capacitance of 22.1 pF
Cylindrical capacitors consist of two conducting cylinders with radii R1 (inner) and R2 (outer) and length L
Capacitance depends on the radii, length, and dielectric constant: C=ln(R2/R1)2πε0εrL
Example: a with R1=2 mm, R2=5 mm, L=10 cm, and εr=4 has a capacitance of 354 pF
Charge and energy storage in capacitors
Capacitors store electric charge when a (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 (U)
Energy stored depends on the capacitance and the applied voltage: U=21CV2
Example: a 10 µF charged to 100 V stores 50 mJ of energy
Capacitors can be connected in series or parallel to achieve desired capacitance values
Series connection: the equivalent capacitance (Ceq) is the reciprocal of the sum of the reciprocals of the individual capacitances: Ceq1=C11+C21+⋯
Parallel connection: the equivalent capacitance is the sum of the individual capacitances: Ceq=C1+C2+⋯
Example: two 10 µF capacitors in series have an equivalent capacitance of 5 µF, while in parallel, their equivalent capacitance is 20 µF
The 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
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 between charges
Biological Applications
Capacitance in biological cell membranes
Cell membranes, composed of a , 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 (Cm) depends on the membrane's surface area (A), thickness (d), and dielectric constant (εr): Cm=dε0εrA
Typical values of membrane capacitance range from 0.5 to 1.3 µF/cm2
Example: a cell with a surface area of 1000 µm2 and a membrane capacitance of 1 µF/cm2 has a total membrane capacitance of 10 pF
Membrane capacitance plays a crucial role in the propagation of electrical signals in neurons
Contributes to the membrane time constant (τm), which determines the rate of change of the membrane potential in response to a current: τm=RmCm, where Rm 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 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