The correct answer is C. 63%.
The time constant of an RC circuit is the amount of time it takes for the current to reach 63% of its final value. This is because the current in an RC circuit is governed by the equation $I(t) = I_0(1 – e^{-t/\tau})$, where $I_0$ is the initial current, $t$ is the time, and $\tau$ is the time constant. When $t = \tau$, $I(t) = I_0(1 – e^{-1}) = 0.6321$, or 63.21% of $I_0$.
Option A is incorrect because 37% is not the percentage of the initial maximum value that the charging current falls to after one time constant.
Option B is incorrect because 42% is not the percentage of the initial maximum value that the charging current falls to after one time constant.
Option D is incorrect because 73% is not the percentage of the initial maximum value that the charging current falls to after one time constant.