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Step-by-Step Solution
Step 1: Identify the quantities and their dimensions
We are given three physical constants:
1. $c$: speed of light, with dimension $[LT^{-1}]$.
2. $G$: universal gravitational constant, with dimension $[M^{-1}L^{3}T^{-2}]$.
3. $\frac{e^2}{4\pi\varepsilon_0}$: its dimension can be analyzed as follows. Since $\frac{e^2}{4\pi\varepsilon_0}$ has the dimension of force × distance² (like an electrostatic potential energy times distance, or Coulomb’s law factor), the dimension is $[ML^3T^{-2}]$.
Step 2: Set up the dimensional equation
We want a quantity of dimension length (i.e., $[L]$) using $c$, $G$, and $\frac{e^2}{4\pi\varepsilon_0}$. Suppose the required quantity (length) is:
$$
L = [c]^x \; [G]^y \; \bigg[\frac{e^2}{4\pi\varepsilon_0}\bigg]^z
$$
Our goal is to find exponents $x$, $y$, and $z$.
Step 3: Express dimensions in symbols
Write the dimension of each quantity in terms of $M$, $L$, and $T$:
$
[c]^x = (L T^{-1})^x = L^x T^{-x},
$
$
[G]^y = (M^{-1}L^3T^{-2})^y = M^{-y} L^{3y} T^{-2y},
$
$
\bigg[\frac{e^2}{4\pi\varepsilon_0}\bigg]^z = (M L^3 T^{-2})^z = M^z L^{3z} T^{-2z}.
$
Hence, the product is:
$
M^{-y} \cdot M^z \times L^{x + 3y + 3z} \times T^{-x -2y -2z}.
$
Step 4: Equate the dimensions to those of length
We require the overall dimension to be $[L^1]$, which implies:
1) For $M$: $-y + z = 0$.
2) For $L$: $x + 3y + 3z = 1$.
3) For $T$: $-x - 2y - 2z = 0$.
Step 5: Solve the dimension equations
From $-y + z = 0$, we get
$
z = y.
$
Now from $-x - 2y - 2z = 0$, substituting $z = y$,
$
-x - 2y - 2y = -x - 4y = 0 \implies x = -4y.
$
From $x + 3y + 3z = 1$ and again substituting $z = y$,
$
-4y + 3y + 3y = -4y + 6y = 2y = 1 \implies y = \frac{1}{2}.
$
Then $z = \frac{1}{2}$ and $x = -4 \cdot \frac{1}{2} = -2$.
Step 6: Construct the final expression
Therefore,
$
x = -2, \quad y = \frac{1}{2}, \quad z = \frac{1}{2}.
$
Substituting back into
$
L = [c]^x [G]^y \bigg[\frac{e^2}{4\pi\varepsilon_0}\bigg]^z,
$
we get:
$
L = \frac{1}{c^2} \left( G \frac{e^2}{4\pi\varepsilon_0} \right)^{\frac{1}{2}}.
$
Step 7: Write the final answer
Thus, the required physical quantity of dimension length is:
$$
\frac{1}{c^2} \sqrt{ G \; \frac{e^2}{4\pi\varepsilon_0} }.
$$
