Question

If the capacitance of a nano capacitor is measured in terms of a unit ${}^{\prime }{\mu }^{\prime }$ made by combining the electronic charge 'e', bohr radius '$a_0$', Planck's constant '$h$' and speed of light '$c$' then which of the following relation is possible?

• A. $\mu =\frac{e^{2}c}{h{a}_0}$
• B. $\mu =\frac{e^{2}h}{c{a}_0}$
• C. $\mu =\frac{hc}{e^2{a}_0}$
• D. $\mu =\frac{e^2{a}_0}{hc}$

Answer 1

The correct option is D $\mu =\frac{e^2{a}_0}{hc}$

$\frac{e^2}{4\pi {ϵ}_{0}hc}$ is a dimensionless constant

Dimension of ${ϵ}_0$ has dimension of $\frac{e^2}{hc}$

Capacitance C has dimension of $\frac{{ϵ}_{0}A}{d}$ i.e $dim\left[{ϵ}_0\right]\left[length\right]$

$\therefore$ capacitance $\mu$ can be expressed in units of ${ϵ}_0\times length$

Bohr's radius $a_0$ has unit of length.

$\therefore \mu =\frac{e^2{a}_0}{hc}$