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Looking at the formula $C = \frac{Q}{U}$ does a high capacitance mean that with a low voltage we can bring more charge onto the capacitor? How is total charge on the capacitor linked with voltage?

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If the potential difference across the plates of the capacitor $U$ is kept constant then the charge stored on the capacitor $Q$ is proprtional to the capacitance $C$, ie $Q\propto C$.

On the other hand if the capacitance $C$ is constant then the charge stored on the capacitor $Q$ is proportional to the potential difference across the plates $U$, ie $Q\propto U$.

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