Step 1: Use the charging equation of the case study:
The passage gives $q = Q\left[1 - e^{-t/RC}\right]$. The power of an exponential must be a pure number.
Step 2: Apply the rule that the exponent has no dimension:
So $t/RC$ has no dimension. This means $[RC] = [t]$.
The product $RC$ is called the time constant for this reason.
Step 3: Write the dimension of time:
Time has the dimension $[M^0L^0T^{1}A^0]$.
Step 4: Cross-check with basic units:
Take $R = 1\,\Omega = 1\,\text{V/A}$ and $C = 1\,\text{F} = 1\,\text{C/V}$. Then $RC = \dfrac{V}{A}\cdot\dfrac{C}{V} = \dfrac{C}{A}$. Since $1\,\text{A} = 1\,\text{C/s}$, this is $\dfrac{C}{C/s} = 1\,\text{s}$.
So the unit of $RC$ is the second, which agrees with the dimension of time.
Step 5: Check the options:
(A) and (C) are the dimensions of $R$ and $C$ separately, so neither can be time. (B) is $[T^{-1}]$, a frequency, the opposite of what is needed. Only (D) has $T^{1}$ with everything else zero.
Final Answer:
$RC$ has the dimension of time, so option (D).
\[ \boxed{[M^0L^0T^{1}A^0]} \]