If L , C and R denote the inductance, capacitance and resistance respectively, the dimensional formula for C…

If L ,  C and R denote the inductance, capacitance and resistance respectively, the dimensional formula for C 2 L R is
  1. [ ML − 2 T − 1 I 0 ]
  2. M 0 L 0 T 3
  3. [ M − 1 L − 2 T 6 I 2 ]
  4. [ M 0 L 0 T 2 I 0 ]

Solution

C 2 L R = C 2 L 2 × R L

As L R and L C  both have the dimensions of time, we get

C 2 L R = M 0 L 0 T 4 × M 0 L 0 T - 1 = M 0 L 0 T 3

~ Alternative Solution: Sure, let's break it down. We know that the dimensional formula of capacitance (C) is $[M^{-1}L^{-2}T^4I^2]$, inductance (L) is $[ML^2T^{-2}I^{-2}]$, and resistance (R) is $[ML^2T^{-3}I^{-2}]$. The question wants the dimensional formula for $C^2LR$. So, we have to multiply the dimensional formula of $C^2$, L, and R. Let's calculate: The dimensional formula for $C^2$ is $[M^{-1}L^{-2}T^4I^2]^2 = [M^{-2}L^{-4}T^8I^4]$. Now, we multiply this by the dimensional formula for L and R: $[M^{-2}L^{-4}T^8I^4] \times [ML^2T^{-2}I^{-2}] \times [ML^2T^{-3}I^{-2}] = [M^{-2+1+1}L^{-4+2+2}T^{8-2-3}I^{4-2-2}] = [M^0L^0T^3I^0]$. So, the correct answer is Option B, which is $[M^0L^0T^3I^0]$.

Asked in: JEE Mains - Units and Dimensions - Test 3

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