Final answer:
To calculate inductance, the time constant formula τ = L/R is used. For the circuit provided, the inductance is calculated to be 43.4 H with a 20-Ω resistor, and it would take approximately 1.0 s to reach 90% of the final current value with a 100-Ω resistor.
Step-by-step explanation:
Calculating Inductance from Circuit Data
To determine the value of inductance from the given data, we use the formula for the time constant (τ) of an RL circuit, which is τ = L/R where L is inductance and R is resistance. The time constant is the time it takes for the current to reach 63.2% of its final value, or alternatively, the time taken to decay to 36.8% of its initial value during discharging. In the data provided, if the current reaches 90% of its final value after 5.0 seconds, we can use the natural logarithm to solve for the time constant as follows:
τ = -t / ln(1 - 0.9)
Using the given value of t = 5.0 s:
τ = -5.0 s / ln(0.1) ≈ 5.0 s / 2.3026 ≈ 2.17 s
Once we know the time constant and the resistance value, we can calculate the inductance (L) by rearranging the time constant formula to L = τ × R. For the resistance R = 20 Ω:
L = 2.17 s × 20 Ω = 43.4 H
If the resistor is replaced with a 100-Ω resistor, the time constant would change accordingly, and using the same natural log relationship as above, the time it would take for the current to reach 90% would be different:
τ' = 43.4 H / 100 Ω = 0.434 s
The time (t') to reach 90% is calculated as follows:
t' = -τ' × ln(1 - 0.9) = -0.434 s × ln(0.1) ≈ 1.0 s