Final answer:
The time constant (τ) in an RC circuit is calculated using τ = RC. Depending on the series or parallel configuration of capacitors and resistors, four combinations with varying time constants are possible. After two RC time constants, a charging capacitor reaches approximately 86.5% of the final voltage, emf.
Step-by-step explanation:
Calculating RC Time Constants
When calculating the RC time constants for a circuit with capacitors and resistors, we need to consider the configurations of the components. Two capacitors with capacitances of 2.00 µF and 7.50 µF can be connected either in series or in parallel, leading to different equivalent capacitances. The same goes for two resistors with resistances of 25.0 kΩ and 100 kΩ. The formula for the time constant (τ) in an RC circuit is given by τ = RC.
In series, the capacitance (Cs) uses the reciprocal sum formula: 1/Cs = 1/C₁ + 1/C₂. In parallel, the capacitance (Cp) is the sum of the individual capacitances: Cp = C₁ + C₂. Similarly, resistance in series (Rs) is the sum of individual resistances: Rs = R₁ + R₂, and in parallel (Rp), it's the reciprocal sum: 1/Rp = 1/R₁ + 1/R₂.
For the given capacitors and resistors, we have four combinations:
- RC time constant with capacitors in series and resistors in series: τss = RsCs
- RC time constant with capacitors in series and resistors in parallel: τsp = RpCs
- RC time constant with capacitors in parallel and resistors in series: τps = RsCp
- RC time constant with capacitors in parallel and resistors in parallel: τpp = RpCp
After two time constants, the voltage on the capacitor will reach approximately 86.5% of the emf, as after one RC time constant (t = RC), the voltage is already at 63.2% of the emf.