If the capacitance of a nanocapacitor is measured in terms of a unit $$u$$, made by combining the electronic charge $$e$$, Bohr radius $$a_0$$, Planck's constant $$h$$ and speed of light $$c$$ then

The capacitance $$ u $$ is to be expressed in terms of the electronic charge $$ e $$, Bohr radius $$ a_0 $$, Planck's constant $$ h $$, and the speed of light $$ c $$. We use dimensional analysis to find the correct expression.

First, recall the dimensions of each quantity in terms of mass (M), length (L), time (T), and current (I):

• Electronic charge $$ e $$ has dimension $$[I T]$$.
• Bohr radius $$ a_0 $$ has dimension $$[L]$$.
• Planck's constant $$ h $$ has dimension $$[M L^2 T^{-1}]$$.
• Speed of light $$ c $$ has dimension $$[L T^{-1}]$$.

The capacitance $$ u $$ has the dimension $$[M^{-1} L^{-2} T^4 I^2]$$. We assume $$ u $$ is proportional to $$ e^a \times a_0^b \times h^c \times c^d $$, so the dimensional equation is:

$$ [M^{-1} L^{-2} T^4 I^2] = [I T]^a \times [L]^b \times [M L^2 T^{-1}]^c \times [L T^{-1}]^d $$

Expanding the right side:

$$ [I T]^a = I^a T^a $$

$$ [L]^b = L^b $$

$$ [M L^2 T^{-1}]^c = M^c L^{2c} T^{-c} $$

$$ [L T^{-1}]^d = L^d T^{-d} $$

Combining these, the right side becomes:

$$ M^c \times L^{b + 2c + d} \times T^{a - c - d} \times I^a $$

Equating dimensions with the left side:

For mass (M): $$ c = -1 $$

For current (I): $$ a = 2 $$

For time (T): $$ a - c - d = 4 $$

Substituting $$ a = 2 $$ and $$ c = -1 $$:

$$ 2 - (-1) - d = 4 $$

$$ 2 + 1 - d = 4 $$

$$ 3 - d = 4 $$

$$ -d = 1 $$

$$ d = -1 $$

For length (L): $$ b + 2c + d = -2 $$

Substituting $$ c = -1 $$ and $$ d = -1 $$:

$$ b + 2(-1) + (-1) = -2 $$

$$ b - 2 - 1 = -2 $$

$$ b - 3 = -2 $$

$$ b = 1 $$

Thus, the exponents are $$ a = 2 $$, $$ b = 1 $$, $$ c = -1 $$, $$ d = -1 $$. The expression for capacitance is:

$$ u \propto e^2 \times a_0^1 \times h^{-1} \times c^{-1} = \frac{e^2 a_0}{h c} $$

Comparing with the options:

• Option A: $$ u = \frac{e^2 a_0}{h c} $$ matches.
• Option B: $$ u = \frac{h c}{e^2 a_0} $$ is the reciprocal.
• Option C: $$ u = \frac{e^2 c}{h a_0} $$ does not match.
• Option D: $$ u = \frac{e^2 h}{c a_0} $$ does not match.

Hence, the correct answer is Option A.