જો કોઈ નેનોકેપેસીટરનું કેપેસીટન્સ વિદ્યુતભાર \(e,\) બોહર ત્રિજ્યા \(a_0,\) પ્લાન્ક અચળાંક \(h\) અને પ્રકાશની ઝડપ \(c\) ના મિશ્રિત એકમ \(u\) થી માપવામાં આવેલ હોય, તો.....

A \(u\, = \,\frac{{{e^2}h}}{{{a_0}}}\)
B \(u\, = \,\frac{{hc}}{{{e^2}{a_0}}}\)
C \(u\, = \,\frac{{{e^2}c}}{{h{a_0}}}\)
D \(u\, = \,\frac{{{e^2}{a_0}}}{{hc}}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(u\, = \,\frac{{{e^2}{a_0}}}{{hc}}\)

Step-by-step Solution

Detailed explanation

The unit of capacitance is \(\frac {Coulomb}{V} = \frac {Coulomb}{\text{work done per unit charge}}\)=\(\frac{\text { Coulomb }}{\frac{ W }{\text { Coulomb }}}=\frac{\text { Coulomb }^{2}}{ kgm ^{2} s ^{-2}}= C ^{2} kg ^{-1} m ^{-2} s ^{2}\) Checking units of each option:…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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