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DC Circuits with Capacitors
A capacitor consists of two charged conductors that are situated opposite to each other and are separated by an insulator. The shape of the conductors is usually chosen so that the electric field is localized in a particular region (parallel-plate, spherical, cylindrical capacitors, etc.)
The capacitance of a capacitor is the ratio of the magnitude of the charge on either of the conductors to the magnitude of the potential difference between the conductors
C = q/U.
The capacitance of a parallel-plate capacitor that consists of two parallel conducting plates, each with area S and separated by a distance d is
C = εε0S/d,
where ε is the dielectric constant of the insulator between the plates, ε0 = 8.85╥10-12 C2/Nm2 is the electric constant, S is the area of each plate, and d is the distance between them.
The equivalent capacitance of a parallel combination equals to the sum of the individual capacitances
C = C1 + C2+ C3 + ... .
The reciprocal of the equivalent capacitance of a serial combination equals to the sum of reciprocals of the individual capacitances
1/C = 1/C1 + 1/C2 + 1/C3 + ... .
A charged capacitor has the potential energy
U = 1/2qV = CV2/2 = q2/2C.
In a circuit with a serial connection of emf ε, an ammeter, a resistor R, and a capacitor C, the initial current depends on time in accordance with the following formula:
I(t) = ε/R?p(-t/RC) .
The product, τ = RC, is called the relaxation time. Upon the passage of time equal to Δt = τ, the current decreases by the reciprocal of e=2,7.
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