Understanding the Dimensions of Physical Quantities

The question asks us to find which physical quantity among the given options has the same dimensions as Planck's constant, often denoted by h. To answer this, we need to determine the dimensions of Planck's constant and then the dimensions of each option and compare them.

Calculating the Dimensions of Planck's Constant (h)

Planck's constant relates the energy (E) of a photon to its frequency (\(\nu\)) through the equation:

\[E = h\nu\]

We can rearrange this equation to find the dimension of h:

\[h = \frac{E}{\nu}\]

Now, let's determine the dimensions of Energy (E) and frequency (\(\nu\)).

• Energy has the dimensions of work, which is force times distance. The dimensions of force are \([MLT^{-2}]\). So, the dimensions of energy are \([MLT^{-2}] \times [L] = [ML^2T^{-2}]\).
• Frequency is the reciprocal of time period, so its dimensions are \([T^{-1}]\).

Substituting these dimensions into the equation for h:

\[[h] = \frac{[ML^2T^{-2}]}{[T^{-1}]} = [ML^2T^{-2}T^{1}] = [ML^2T^{-1}]\]

So, the dimensions of Planck's constant h are \([ML^2T^{-1}]\).

Determining Dimensions of the Options

Let's find the dimensions for each given option:

1. Linear momentum (p): Linear momentum is defined as mass times velocity (\(p = mv\)).
   • Mass dimension: \([M]\)
   • Velocity dimension: \([LT^{-1}]\)
   • Dimension of linear momentum: \([M] \times [LT^{-1}] = [MLT^{-1}]\).
   This is \([MLT^{-1}]\), which is different from \([ML^2T^{-1}]\).
2. Angular momentum (L): Angular momentum is defined as the product of moment of inertia (I) and angular velocity (\(\omega\)), or alternatively, as the cross product of position vector (r) and linear momentum (p) (\(L = r \times p\)). Using the latter definition is often simpler for dimensions.
   • Position vector dimension: \([L]\)
   • Linear momentum dimension: \([MLT^{-1}]\) (as calculated above)
   • Dimension of angular momentum: \([L] \times [MLT^{-1}] = [ML^2T^{-1}]\).
   This is \([ML^2T^{-1}]\), which is the same as the dimensions of Planck's constant h.
3. Force (F): Force is defined as mass times acceleration (\(F = ma\)).
   • Mass dimension: \([M]\)
   • Acceleration dimension: \([LT^{-2}]\)
   • Dimension of force: \([M] \times [LT^{-2}] = [MLT^{-2}]\).
   This is \([MLT^{-2}]\), which is different from \([ML^2T^{-1}]\).
4. Potential energy: Potential energy is a form of energy. As calculated earlier for Energy (E), the dimensions are \([ML^2T^{-2}]\).
   • Dimension of potential energy: \([ML^2T^{-2}]\).
   This is \([ML^2T^{-2}]\), which is different from \([ML^2T^{-1}]\).

Comparison and Conclusion

Comparing the dimensions we calculated:

• Planck's constant h: \([ML^2T^{-1}]\)
• Linear momentum: \([MLT^{-1}]\)
• Angular momentum: \([ML^2T^{-1}]\)
• Force: \([MLT^{-2}]\)
• Potential energy: \([ML^2T^{-2}]\)

Only angular momentum has the same dimensions as Planck's constant.

Therefore, Angular momentum has the same dimensions as that of Planck’s constant h.

Revision Table: Dimensions of Common Physical Quantities

Additional Information on Planck's Constant and Angular Momentum

Planck's constant (h) is a fundamental constant in quantum mechanics. It represents the smallest unit of "action," which has the dimensions of energy multiplied by time, or momentum multiplied by distance. This is exactly the dimension of angular momentum.

The similarity in dimensions between Planck's constant and angular momentum is not a coincidence. In quantum mechanics, angular momentum is quantized, meaning it can only take on discrete values that are integer or half-integer multiples of a fundamental unit related to Planck's constant (\(\hbar = h/2\pi\), where \(\hbar\) is the reduced Planck constant). This fundamental unit \(\hbar\) has the same dimensions as h.

Understanding the dimensions of physical quantities is a powerful tool in physics. It allows us to check the consistency of equations and provides insight into the fundamental nature of physical laws, even hinting at connections between seemingly different concepts like energy-frequency (Planck's constant) and rotational motion (angular momentum).