Question

The Entropy ($S$) of a black hole can be written as $S = \beta k_B A$, where $k_B$ is the Boltzmann constant and $A$ is the area of the black hole. Then $\beta$ has dimension of

• A. L2
• B. ML2T-1
• C. L-2
• D. dimensionless

Answer 1

The Correct Option is A

The entropy \( S \) of a black hole is given by the equation: \[ S = \beta k_B A, \] where \( k_B \) is the Boltzmann constant and \( A \) is the area of the black hole. The units of \( k_B \) are: \[ [k_B] = \text{J/K} = \frac{\text{ML}^2}{\text{T}^2 \text{K}}. \] The area \( A \) has the dimensions of \( \text{L}^2 \) (since area is measured in square units of length). Thus, the dimensions of \( S \) (entropy) are: \[ [S] = \frac{\text{ML}^2}{\text{T}^2 \text{K}} \times \text{L}^2 = \frac{\text{M L}^4}{\text{T}^2 \text{K}}. \] Since entropy \( S \) is dimensionless (as it represents disorder or randomness), the dimensions of \( \beta \) must cancel out the units of \( k_B \) and \( A \). Therefore, \( \beta \) must have the dimensions of \( \text{L}^2 \). Hence, the dimension of \( \beta \) is: \[ \boxed{\text{L}^2}. \]