Question: Dimensions of CR are those of: [PMT/NEET-1999]
a. Frequency
b. Energy
c. Time period
d. Current

Answer: (c) Time period. The product $CR$ has the dimension of time (seconds).

Dimensional proof

• Capacitance $C$: $[C] = M^{-1}L^{-2}T^{4}I^{2}$
• Resistance $R$: $[R] = M L^{2}T^{-3}I^{-2}$
• Product:
  $
  [CR] = \big(M^{-1}L^{-2}T^{4}I^{2}\big)\big(M L^{2}T^{-3}I^{-2}\big) = T
  $
  Hence $CR$ has the dimensions of time.

Physical insight (RC circuits)

• In an RC circuit, the charging/discharging law is $V_C(t) = V_0\big(1 – e^{-t/RC}\big)$ or $q(t) = Q e^{-t/RC}$.
• The quantity $\tau = RC$ is the time constant, so $RC$ must have units of seconds. Since multiplication is commutative, $CR$ has the same dimension.

Option check

• Frequency: $[T^{-1}]$ → incorrect
• Energy: $[ML^{2}T^{-2}]$ → incorrect
• Time period: $[T]$ → correct
• Current: $[I]$ → incorrect