Final answer:
The RC time constant is calculated by multiplying the resistance (R) and capacitance (C) in a series circuit. There are four different RC time constants possible from different series and parallel combinations of a 2.00- and a 7.50-μF capacitor and a 25.0- and a 100-k ohm resistor. After two time constants, a capacitor reaches approximately 86.5% of the final voltage.
Step-by-step explanation:
The student's question relates to the calculation of RC time constants for certain combinations of capacitors and resistors. The RC time constant (τ) is the product of the resistance (R) and the capacitance (C) in a series circuit, which defines the time it takes for the voltage across the capacitor to either charge up to 63.2% of its maximum value or discharge to 36.8% of its initial value. When we have a 2.00- and a 7.50-μF capacitor and a 25.0- and a 100-k ohm resistor, these components can be combined in series or parallel to create different capacitance and resistance values, resulting in four possible RC time constants.
For a parallel combination of capacitors, the total capacitance (Ctotal) is the sum of the individual capacitances. When capacitors are in series, the inverse of the total capacitance is the sum of the inverses of the individual capacitances. For resistors, it's the opposite: the total resistance for a series combination is the sum of individual resistances, while for a parallel combination, the inverse of the total resistance is the sum of the inverses of the individual resistances. Once we have the total R and C for each scenario, we can calculate the RC time constant for each.
Furthermore, after two time constants, a capacitor will have charged up to a certain percentage of the final voltage (emf). This is known as the exponential charging curve of a capacitor. Specifically, after two RC time constants, the capacitor will have reached about 86.5% of the final voltage.