The time constant of the circuit is the product of the resistance and the inductance, and it represents the time it takes for the current to reach 63.2% of its maximum value when changing from 0 to its final value.
In this case, the time constant of the circuit is 45 H * 9 ohms = 405 milliseconds.
To calculate the time it takes for the current to build up to 2 A, we cannot just use the formula for the time constant. Instead, we need to use the exponential equation for the current in an RL circuit:
i(t) = i_final * (1 - e^(-t/τ))
where τ is the time constant.
Setting i(t) = 2 A, we can solve for t:
2 = 2 * (1 - e^(-t/405 ms))
e^(-t/405 ms) = 0.5
-t/405 ms = ln(0.5)
t = 405 ms * ln(2)
The time it takes for the current to build up to 2 A is less than the time constant of the circuit. This means that the circuit will reach 63.2% of its maximum current within 405 ms, but it will take longer to reach 2 A.