Question

If \( G \) is the Gravitational constant and \( h \) is Planck's constant, then the dimension of \( G \) is:

• A. \( [M^{-2} L^1 T^{-1} h^2] \)
• B. \( [M^{-2} L^3 T^{-1} h^1] \)
• C. \( [M^{-1} L T^{-3} h^3] \)
• D. \( [M^{-2} L T^{-1}] \)

Hint:

For dimensions of physical constants, use their defining equations and substitute the dimensions of the variables involved.

Answer 1

The Correct Option is B

Step 1: Understand the given constants.
We are given two physical constants:
- \( G \) is the gravitational constant.
- \( h \) is Planck's constant.
The dimensional formula for each of these constants can be derived from their respective physical equations.
Step 2: Gravitational constant \( G \).
From Newton's law of gravitation, the formula for gravitational force is: \[ F = \frac{G m_1 m_2}{r^2} \] The dimensions of \( F \) (force) are \( [M L T^{-2}] \), and the dimensions of \( m_1, m_2 \) are \( [M] \) (mass), and \( r \) (distance) is \( [L] \). Rearranging the formula to solve for \( G \): \[ G = \frac{F r^2}{m_1 m_2} \] Substitute the dimensions: \[ G = \frac{[M L T^{-2}] [L]^2}{[M]^2} = [M^{-1} L^3 T^{-2}] \]
Step 3: Planck's constant \( h \).
The dimension of Planck's constant \( h \) is derived from the equation \( E = h f \), where \( E \) is energy and \( f \) is frequency. The dimension of \( E \) is \( [M L^2 T^{-2}] \), and the dimension of \( f \) is \( [T^{-1}] \). Thus, the dimension of \( h \) is: \[ [h] = \frac{[M L^2 T^{-2}]}{[T^{-1}]} = [M L^2 T^{-1}] \]
Step 4: Combine the dimensions.
Now, using the dimensions of \( G \) and \( h \), we find the dimension of \( G \) as: \[ [G] = [M^{-1} L^3 T^{-2}] \times [M L^2 T^{-1}] = [M^{-2} L^3 T^{-1} h^1] \] Final Answer: \( [M^{-2} L^3 T^{-1} h^1] \)