The dimensions of Planck’s constant is same as that of

A. Angular momentum

B. Linear momentum

C. Work

D. Coefficient of viscosity

To determine which quantity has the same dimensions as Planck's constant, we first need to understand what Planck's constant represents and its dimensional formula. Planck's constant (denoted as h) is a fundamental constant in quantum mechanics that relates the energy of a photon to its frequency. The formula for energy (E) is given by E = hν, where ν (nu) is the frequency. From this relationship, we can derive the dimensions of Planck's constant.

Breaking Down Planck's Constant

Energy has the dimension of mass times acceleration (force), which can be expressed as:

• Energy (E) = Mass (M) × Acceleration (L/T²) = M L² T⁻²

Since Planck's constant relates energy to frequency, we can express frequency (ν) as the inverse of time:

• Frequency (ν) = 1/Time (T) = T⁻¹

Now, substituting frequency into the equation for energy gives us:

• h = E/ν = (M L² T⁻²) / (T⁻¹) = M L² T⁻¹

Comparing with Other Quantities

Now that we have established the dimensions of Planck's constant as M L² T⁻¹, let's compare this with the dimensions of the options provided:

• A. Angular momentum: The dimension of angular momentum is given by the product of moment of inertia (M L²) and angular velocity (T⁻¹), which results in M L² T⁻¹. This matches Planck's constant.
• B. Linear momentum: The dimension of linear momentum is mass times velocity (M L/T), which simplifies to M L T⁻. This does not match.
• C. Work: The dimension of work is force times distance (M L² T⁻²), which does not match either.
• D. Coefficient of viscosity: The dimension of viscosity is M L⁻¹ T⁻², which is also different.

Final Thoughts

From this analysis, we can conclude that the dimensions of Planck's constant are indeed the same as those of angular momentum. Therefore, the correct answer is:

A. Angular momentum

This relationship highlights the deep connections between classical mechanics and quantum mechanics, illustrating how fundamental constants bridge different realms of physics.