Question

Which of the following dimensions will be the same as that of time?

• A. $\frac{L}{R}$
• B. $\frac{C}{L}$
• C. $LC$
• D. $\frac{R}{L}$

Answer 1

The Correct Option is A

To determine which of the given options has the same dimension as that of time, we need to analyze the dimensional formulae involved.

1. First, consider each of the elements involved:
   • L represents inductance with a dimensional formula of [M^1 L^2 T^{-2} I^{-2}], where M is mass, L is length, T is time, and I is current.
   • R represents resistance with a dimensional formula of [M^1 L^2 T^{-3} I^{-2}].
   • C represents capacitance with a dimensional formula of [M^{-1} L^{-2} T^4 I^2].
2. Now, evaluate each option:
   • Option A: \frac{L}{R}
     The dimensional formula is \frac{[M^1 L^2 T^{-2} I^{-2}]}{[M^1 L^2 T^{-3} I^{-2}]} = [T]. Therefore, the dimension is the same as that of time.
   • Option B: \frac{C}{L}
     The dimensional formula is \frac{[M^{-1} L^{-2} T^4 I^2]}{[M^1 L^2 T^{-2} I^{-2}]} = [M^{-2} L^{-4} T^{6} I^4]. The dimensions do not match those of time.
   • Option C: LC
     The dimensional formula is [M^1 L^2 T^{-2} I^{-2}] \times [M^{-1} L^{-2} T^4 I^2] = [T^2]. The dimension represents time squared, not time.
   • Option D: \frac{R}{L}
     The dimensional formula is \frac{[M^1 L^2 T^{-3} I^{-2}]}{[M^1 L^2 T^{-2} I^{-2}]} = [T^{-1}]. The dimensions do not match those of time.
3. From the above analysis, \frac{L}{R} has the dimension of time [T].

Thus, the correct answer is \frac{L}{R}.