The voltage across a capacitor that is charging from a direct current supply will increase what percentage for each time constant?

The correct answer pertains to the behavior of a capacitor in an RC (resistor-capacitor) charging circuit when connected to a direct current (DC) supply. The concept of time constant, represented by the Greek letter tau (τ), is crucial here. The time constant is defined as the product of the resistance (R) and the capacitance (C) in the circuit (τ = R × C).

When a capacitor is charging, the voltage across it does not increase instantaneously. Instead, it follows an exponential growth curve. Specifically, after one time constant (τ), the voltage across the capacitor will reach approximately 63.2% of the maximum voltage (the voltage of the DC supply). This percentage is derived from the mathematical relationship governing the charging of a capacitor, which is expressed as:

[ V(t) = V_{max} \left(1 - e^{-t/τ}\right) ]

At time t = τ, the equation simplifies to show that approximately 63.2% of the full supply voltage is reached.

Understanding this behavior is fundamental in electrical engineering, particularly in the analysis of transient responses in circuits. This characteristic is essential for designing and troubleshooting circuits that involve capacitors, ensuring that they function as