Match List - I with List - II (where h is Planck's constant, G is gravitational constant and c is speed of light):
- A
- B
- C
- D
The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept:
We use dimensional analysis to match units to their expressions in terms of fundamental constants $h, G, c$.
Step 2: Detailed Explanation:
Let $[L] = h^a G^b c^d$.
Dimensions: $[h] = ML^2T^{-1}, [G] = M^{-1}L^3T^{-2}, [c] = LT^{-1}$.
For Length: $M^0 L^1 T^0 = (ML^2T^{-1})^a (M^{-1}L^3T^{-2})^b (LT^{-1})^d$
$a - b = 0 \implies a = b$.
$2a + 3b + d = 1 \implies 5a + d = 1$.
$-a - 2b - d = 0 \implies -3a - d = 0 \implies d = -3a$.
$5a - 3a = 1 \implies 2a = 1 \implies a = 1/2$.
So $b = 1/2, d = -3/2$.
Expression: $\sqrt{\frac{hG}{c^3}}$. (Matches A-I).
Similarly for Time: $a=1/2, b=1/2, d=-5/2 \implies \sqrt{\frac{hG}{c^5}}$. (Matches B-II).
For Mass: $a=1/2, b=-1/2, d=1/2 \implies \sqrt{\frac{hc}{G}}$. (Matches C-III).
Step 3: Final Answer:
The correct matching is A-I, B-II, C-III.
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