Final answer:
The characteristic time constant for the given RL circuit is 6.25 ms. After 12.5 ms of connecting a 25.0 mH inductor to a 12.0 V battery, the current would be approximately 2.68 A.
Step-by-step explanation:
Characteristic Time Constant and Current in an RL Circuit
The characteristic time constant (τ) of an RL circuit is determined by the formula τ = L/R, where L is the inductance and R is the resistance.
For a 25.0 mH inductor and a 4.00 Ω resistance, the time constant is
τ = (25.0 x 10^-3 H) / (4.00 Ω)
= 6.25 ms.
When the inductor is connected to a 12.0 V battery, the incremental current at a given time t can be found using the equation I(t) = Io(1 - e^{-t/τ}), where Io is the final current.
The final current Io equals the voltage divided by resistance (Io = V/R),
so,
Io = (12.0 V) / (4.00 Ω) =
3.00 A.
After 12.5 ms, the current is I(12.5 ms) = 3.00 A(1 - e^{-12.5 ms / 6.25 ms}), which equals approximately 2.68 A.