Question

$A, B, C$ and $D$ are four different physical quantities having different dimensions. None of them is dimensionless. But we know that the equation $AD = C$ ln $(BD)$ holds true. Then which of the combination is not a meaningful quantity ?

• A. $A^2 - B^2 C^2$
• B. $\frac{(A - C)}{D}$
• C. $\frac{A}{B} - C$
• D. $\frac{C}{BD} - \frac{AD^2}{C}$

Answer 1

The Correct Option is B

To determine which of the provided combinations is not a meaningful quantity, we need to analyze the given equation and options. The equation is:

AD = C \ln (BD)

Let's break down the dimensions based on this equation:

1. AD must have the same dimensions as C \ln (BD). Since \ln (BD) is dimensionless, it implies that C must have the same dimensions as AD.
2. Therefore, the dimensions of C are equal to those of AD.

Now, let's evaluate the options given:

• Option 1: A^2 - B^2 C^2
  • A^2: Dimensions of A squared.
  • B^2 C^2: Dimensions of B squared multiplied by dimensions of C squared.
• Option 2: \frac{(A-C)}{D}
  • A - C: This operation is only dimensionally valid if A and C have the same dimensions.
  • \frac{(A-C)}{D}: Subtracting two quantities of the same dimension gives zero if they are not actually equal in value, making this expression dimensionally invalid.
• Option 3: \frac{A}{B} - C
  • \frac{A}{B}: Dimensionally valid operation.
  • The subtraction is valid if dimensions are matched, but here they do not mention dimensional compatibility directly.
• Option 4: \frac{C}{BD} - \frac{AD^2}{C}
  • Both terms have clear dimensional validity under operations described above, following the dimension analysis.

From the above analysis, Option 2: \frac{(A - C)}{D} is not a meaningful quantity because it involves subtracting quantities in the numerator that, while dimensionally aligned, result in a dimensionless operation with the denominator D, essentially invalidating the operation.