Step 1: Understanding the Concept:
Dimensional analysis is a fundamental tool in physics used to ensure that equations are physically consistent.
Every physical quantity can be expressed in terms of base dimensions: Mass (\(M\)), Length (\(L\)), Time (\(T\)), and Current (\(A\)).
Quantities like energy, frequency, and potential have standard dimensional formulas that can be derived from basic physical laws.
Step 2: Detailed Explanation:
A. Planck's constant (\(h\)):
From Einstein's photoelectric equation or the energy of a photon: \(E = h\nu\).
Energy (\(E\)) has dimensions \([ML^2T^{-2}]\).
Frequency (\(\nu\)) has dimensions \([T^{-1}]\).
Dimensions of \(h = \frac{[E]}{[\nu]} = \frac{[ML^2T^{-2}]}{[T^{-1}]} = [ML^2T^{-1}]\).
This matches with III.
B. Stopping potential (\(V_0\)):
Potential (\(V\)) is defined as work done per unit charge (\(W/q\)).
Work has dimensions \([ML^2T^{-2}]\).
Charge (\(q = I \cdot t\)) has dimensions \([AT]\).
Dimensions of \(V_0 = \frac{[ML^2T^{-2}]}{[AT]} = [ML^2T^{-3}A^{-1}]\).
This matches with IV.
C. Work function (\(\Phi\)):
Work function is the minimum energy required to remove an electron from a metal surface.
Since it is a form of energy, it has the same dimensions as work or kinetic energy.
Dimensions = \([ML^2T^{-2}]\).
This matches with I.
D. Threshold frequency (\(\nu_0\)):
As the name suggests, it is a frequency.
Frequency is defined as cycles per second, having dimensions of inverse time.
Dimensions = \([T^{-1}]\).
This matches with II.
Step 3: Final Answer:
The correct mapping is: A-III, B-IV, C-I, D-II.