A Hypothetical RC Circuit

Electricity and Magnetism Level 3

\[R(a) = \sum_{n=1}^{\infty} \frac{1}{a^n} = \frac{1}{a-1} \qquad \qquad C(a) = \sum_{n=0}^{\infty} \frac{1}{a^n} = \frac{a}{a-1} \]

An RC ciruit consisting of a multitude of resistors (\(R\)) in series followed by a multitude of capacitors (\(C\)) in parallel is prepared such that the resistance and capacitance can be calculated with the series written above. At \(a= 3\), calculate the voltage (\(V\)) of the capacitor when it has charged for \(t = 2.5 \space \text{seconds}\), where the circuit's power supply voltage (\(V_{\circ}\)) is \(14 \space \text{volts}\).

Round your answer to three significant figures.

Note: \(V(t) = V_{\circ}(1-e^{N}) \) where \( N = \frac{\text{-t}}{\text{RC}} \).


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