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Step-by-Step Solution
Step 1: Identify the Desired Physical Quantity
The question asks for the dimensions of the stopping potential, denoted by $V_{0}$, in terms of Planck's constant ($h$), speed of light ($c$), gravitational constant ($G$), and the ampere ($A$). By definition, a potential (in SI units) has the dimension:
$[V_{0}] = M^{1}L^{2}T^{-3}A^{-1}.$
Step 2: Write Dimensions of the Given Constants
Planck's Constant, $h$:
$
[h] = M^{1}L^{2}T^{-1}.
$
Speed of Light, $c$:
$
[c] = L^{1}T^{-1}.
$
Gravitational Constant, $G$:
$
[G] = M^{-1}L^{3}T^{-2}.
$
Ampere, $A$:
$
[I] = A.
$
Step 3: Assume General Dimensional Dependence
Let the stopping potential depend on these constants as follows:
$
V_{0} \propto h^{P}\,c^{Q}\,G^{R}\,A^{S}.
$
This means dimensionally:
$[V_{0}] = [h]^{P}\,[c]^{Q}\,[G]^{R}\,[A]^{S}.$
Step 4: Express Dimensions in Terms of M, L, T, and A
Substitute the dimensions:
$
\begin{aligned}
[V_{0}] &= M^{1}L^{2}T^{-3}A^{-1},\\[6pt]
[h]^{P} &= \bigl(M^{1}L^{2}T^{-1}\bigr)^{P} = M^{P}L^{2P}T^{-P},\\
[c]^{Q} &= \bigl(L^{1}T^{-1}\bigr)^{Q} = L^{Q}T^{-Q},\\
[G]^{R} &= \bigl(M^{-1}L^{3}T^{-2}\bigr)^{R} = M^{-R}L^{3R}T^{-2R},\\
[A]^{S} &= A^{S}.
\end{aligned}
$
Hence, multiplying these dimensions gives:
$
M^{P - R} \, L^{2P + Q + 3R} \, T^{-P - Q - 2R} \, A^{S}.
$
Step 5: Equate Exponents with the Dimensions of $V_{0}$
We match each exponent of $M, L, T,$ and $A$ with those in $[V_{0}] = M^{1}L^{2}T^{-3}A^{-1}.$ This yields the system of equations:
$
\begin{aligned}
P - R &= 1,\\
2P + Q + 3R &= 2,\\
-\,P - Q - 2R &= -3, \\
S &= -1.
\end{aligned}
$
Step 6: Solve for the Exponents P, Q, R, and S
From $S = -1$ directly. Next, solve for $P, Q,$ and $R$:
From $P - R = 1$, assume $P = R + 1$.
Use $-\,P - Q - 2R = -3$ and $2P + Q + 3R = 2$ to solve simultaneously.
On solving, we get:
$
P = 0, \quad Q = 5, \quad R = -1, \quad S = -1.
$
Step 7: Write the Final Dimensional Form
Substitute these values back into the expression for $V_{0}$:
$
V_{0} \propto h^{0} \, c^{5} \, G^{-1} \, A^{-1}.
$
This corresponds to the dimension:
$
\displaystyle h^{0}\, c^{5}\, G^{-1}\, A^{-1},
$
which matches the correct choice given in the options.
