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Because the voltage across the capacitor is initially zero, making the voltage across the resistor maximal. As the capacitor charges, the voltage across the resistor decreases, with an accompanying reduction in current. At maximum charge, the voltage across the resistor is zero, and thus there is no current.

The equation of a capacitor is ...

dv/dt = i/c

... meaning that the rate of change of voltage, in volts per second, is proportional to current, in amperes, and inversely proportional to capacitance, in farads.

Set this up into a series charging circuit, replacing the i term with (V-Vt) / R, solve the differential equation, and you get ...

Vt = VSource (1 - e-t/RC)

.... where Vt is voltage at some time t, and RC is the time constant, resistance times capacitance.

The is the DC state answer. AC circuits are a whole other issue, and require different analysis, though the fundamental equation remains the same.

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