Planck's constant has the dimensions of :a)forceb)energyc)linear momentumd)angular momentumCorrect answer is option 'D'. Can you explain this answer?

Question

Accepted Answer

We know that E = hv , where E is energy and v is frequency.
Thus we get h = E/v
And [h] = [E/v] = ML²T^-2 / T^-1
= ML²T^-1
[P] = MLT^-1
[F] = MLT^-2
[Angular momentum] = [P x r] = ML²T^-1

Answer

We know that E=hν
where dimensions of Energy, E are ML2T−2 and dimensions of frequency are inverse of Time Period, i.e. T−1
Thus, ML2T−2=[h]T−1
or, [h]=ML2T−1
Now, angular momentum, L=Iω
where units of omega are inverse of time (rad/sec, where radian is dimensionless).
Units of I are ML2 (because I is of the form kMx2)
Thus [L]=ML2T−1

Answer

Understanding Planck's Constant
Planck's constant (h) is a fundamental quantity in quantum mechanics, linking energy and frequency of photons. Its dimensions reveal its relationship to angular momentum.
Dimensional Analysis
- The dimensions of Planck's constant can be understood through the equation:
- E = hν, where E is energy, h is Planck’s constant, and ν is frequency.
- Rearranging gives:
- h = E/ν
Energy and Frequency
- Energy (E) has the dimension of mass multiplied by the square of length divided by time squared (ML^2/T^2).
- Frequency (ν) has the dimension of inverse time (1/T).
- Thus, the dimension of Planck's constant becomes:
- [h] = (ML^2/T^2) / (1/T) = ML^2/T
Angular Momentum Relation
- Angular momentum (L) has the dimension of mass times length squared per time (ML^2/T), which is identical to the derived dimension of Planck's constant.
- Therefore, Planck's constant has the same dimensions as angular momentum.
Conclusion
- The correct answer is option 'D' because Planck's constant fundamentally relates to angular momentum in quantum mechanics.
- This connection underpins many phenomena in physics, including the quantization of energy levels in atoms and the behavior of particles at the quantum level.

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