Question:
If the dimensions of length are expressed as \( G^x C^y h^z \), where \( G \), \( C \), and \( h \) are the universal gravitational constant, speed of light, and Planck's constant respectively, then:
If the dimensions of length are expressed as \( G^x C^y h^z \), where \( G \), \( C \), and \( h \) are the universal gravitational constant, speed of light, and Planck's constant respectively, then:
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Write the dimensions of \( G \), \( c \) and \( h \) in terms of M, L and T. Then match the powers of M, L and T on both sides of \( L = G^x c^y h^z \).
Updated On: Nov 5, 2026
- \( x = \frac{1}{2}, y = \frac{1}{2} \)
- \( x = \frac{1}{2}, z = \frac{1}{2} \)
- \( y = \frac{1}{2}, z = \frac{3}{2} \)
- \( y = -\frac{3}{2}, z = \frac{1}{2} \)
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The Correct Option is B, D
Solution and Explanation
Step 1: Understanding the Question:
We want a combination \( G^x c^y h^z \) that has the dimension of length. We find \( x \), \( y \) and \( z \), then check each option.
Step 2: Dimensions of the constants:
\[ [G] = M^{-1}L^{3}T^{-2}, \qquad [c] = LT^{-1}, \qquad [h] = ML^{2}T^{-1} \]
The constant \( \hbar \) has the same dimension as \( h \).
Step 3: Match the powers:
We need \( M^0 L^1 T^0 \) on the left side. Writing the powers of each base gives three equations.
For M: \( -x + z = 0 \), so \( z = x \).
For L: \( 3x + y + 2z = 1 \).
For T: \( -2x - y - z = 0 \).
Step 4: Solve:
Put \( z = x \) in the other two equations.
L gives \( 5x + y = 1 \). T gives \( 3x + y = 0 \), so \( y = -3x \).
Subtract: \( 2x = 1 \), so \( x = \tfrac12 \). Then \( y = -\tfrac32 \) and \( z = \tfrac12 \).
\[ L \sim G^{1/2}\, c^{-3/2}\, h^{1/2} \]
Step 5: Check the options:
(A) says \( y = \tfrac12 \), but \( y = -\tfrac32 \). So (A) is FALSE.
(B) says \( x = \tfrac12, z = \tfrac12 \). This matches. So (B) is TRUE.
(C) says \( y = \tfrac12, z = \tfrac32 \). This does not match. So (C) is FALSE.
(D) says \( y = -\tfrac32, z = \tfrac12 \). This matches. So (D) is TRUE.
Final Answer:
Options (B) and (D) are correct. \[ \boxed{\text{(B) and (D)}} \]
We want a combination \( G^x c^y h^z \) that has the dimension of length. We find \( x \), \( y \) and \( z \), then check each option.
Step 2: Dimensions of the constants:
\[ [G] = M^{-1}L^{3}T^{-2}, \qquad [c] = LT^{-1}, \qquad [h] = ML^{2}T^{-1} \]
The constant \( \hbar \) has the same dimension as \( h \).
Step 3: Match the powers:
We need \( M^0 L^1 T^0 \) on the left side. Writing the powers of each base gives three equations.
For M: \( -x + z = 0 \), so \( z = x \).
For L: \( 3x + y + 2z = 1 \).
For T: \( -2x - y - z = 0 \).
Step 4: Solve:
Put \( z = x \) in the other two equations.
L gives \( 5x + y = 1 \). T gives \( 3x + y = 0 \), so \( y = -3x \).
Subtract: \( 2x = 1 \), so \( x = \tfrac12 \). Then \( y = -\tfrac32 \) and \( z = \tfrac12 \).
\[ L \sim G^{1/2}\, c^{-3/2}\, h^{1/2} \]
Step 5: Check the options:
(A) says \( y = \tfrac12 \), but \( y = -\tfrac32 \). So (A) is FALSE.
(B) says \( x = \tfrac12, z = \tfrac12 \). This matches. So (B) is TRUE.
(C) says \( y = \tfrac12, z = \tfrac32 \). This does not match. So (C) is FALSE.
(D) says \( y = -\tfrac32, z = \tfrac12 \). This matches. So (D) is TRUE.
Final Answer:
Options (B) and (D) are correct. \[ \boxed{\text{(B) and (D)}} \]
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