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So in calculating the voltage across a capacitor, the voltage is equal to the amount of current that has charge (current) that has built up on one side of the capacitor. A higher capacitance results in a larger current for the same rate of voltage change. The voltage across a capacitor can not change instantaneously so the initial voltage across the capacitor is 0

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The capacitor then charges to the value of the input voltage and current stops flowing. The current flowing through a capacitor is directly proportional to the capacitance and the rate of change of voltage This article describes the theory behind charging a capacitor

The page also shows the derivation for the expression of voltage and current during charging of a capacitor.

To determine the change in capacitor voltage and inductor current, it's often necessary to utilize the fundamental relationships between voltage (v) and current (i) for capacitors and inductors, along with kirchhoff's current law (kcl) and kirchhoff's voltage law (kvl). As the capacitor charges, the voltage across the capacitor increases and the current through the circuit gradually decrease For an uncharged capacitor, the current through the circuit will be maximum at the instant of switching. To express the voltage across the capacitor in terms of the current, you integrate the preceding equation as follows

The second term in this equation is the initial voltage across the capacitor at time t = 0