Question

If L, C and R are the self-inductance, capacitance and resistance, respectively, which of the following does not have the dimension of time?

• A. \(RC\)
• B. \(\frac{L}{R}\)
• C. \(\sqrt{LC}\)
• D. \(\frac{L}{C}\)

Answer 1

The Correct Option is D

The problem requires determining which expression among the given options does not have the dimension of time. We will analyze the dimensions of each option:

1. Option 1: RC

   Resistance (R) has dimensions of ML^2T^{-3}I^{-2} and Capacitance (C) has dimensions of M^{-1}L^{-2}T^4I^2. Therefore, the product RC has dimensions:

   (ML^2T^{-3}I^{-2})(M^{-1}L^{-2}T^4I^2) = T.

   This indicates RC has dimensions of time.

2. Option 2: \frac{L}{R}

   Inductance (L) has dimensions ML^2T^{-2}I^{-2} and Resistance (R) has dimensions ML^2T^{-3}I^{-2}. Thus, the ratio \frac{L}{R} has dimensions:

   \frac{ML^2T^{-2}I^{-2}}{ML^2T^{-3}I^{-2}} = T.

   This indicates \frac{L}{R} has dimensions of time.

3. Option 3: \sqrt{LC}

   Inductance (L) and Capacitance (C) gives:

   LC = (ML^2T^{-2}I^{-2})(M^{-1}L^{-2}T^4I^2) = T^2.

   Thus, \sqrt{LC} = T.

   This indicates \sqrt{LC} has dimensions of time.

4. Option 4: \frac{L}{C}

   Substituting the dimensions for Inductance (L) and Capacitance (C):

   \frac{L}{C} = \frac{ML^2T^{-2}I^{-2}}{M^{-1}L^{-2}T^4I^2} = M^2L^4T^{-6}I^{-4}.

   Clearly, \frac{L}{C} does not have dimensions of time.

The dimension of time is evident in options 1, 2, and 3, but option 4 does not have these dimensions. Thus, Option 4: \frac{L}{C} is the correct answer.