JEE Mains · Physics · STD 12 - 12. atoms

A small particle of mass \(m\) moves in such a way that its potential energy \(U=\frac{1}{2} m \omega^2 r ^2\) where \(\omega\) is constant and \(r\) is the distance of the particle from origin. Assuming Bohr's quantization of momentum and circular orbit, the radius of \(n^{\text {th }}\) orbit will be proportional to.

A \(\sqrt{n}\)
B \(n\)
C \(n^2\)
D \(\frac{1}{n}\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\sqrt{n}\)

Step-by-step Solution

Detailed explanation

\(U=\frac{1}{2} m \omega^2 r^2\) \(F=-\frac{d v}{d r}=-m \omega^2 r\) Now \(m \omega^2 r=\frac{m v^2}{r} \Rightarrow v=\omega v^{\prime}\) \(m vr=\frac{n h}{2 \pi} \ldots (ii)\) From \((i)\) and \((ii)\) \(m \omega r^2=\frac{n h}{2 \pi}\) \(\Rightarrow r \propto \sqrt{n}\)

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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The kinetic energy of an electron, \(\alpha\)-particle and a proton are given as \(4K,2K\) and \(K\) respectively. The de-Broglie wavelength associated with electron ( \(\lambda e), \alpha\)-particle \((\lambda \alpha)\) and the proton \((\lambda p)\) are as follows:

JEE Mains 2023 Medium

Match the \(\frac{C _{ P }}{ C _{ v }}\) ratio for ideal gases with different type of molecules

Moleculae type	\(\frac{C _{ P }}{ C _{ v }}\)
\((A)\) Monoatomic	\((I)\) \(\frac{7}{ 5}\)
\((B)\) Diatomic rigid molecules	\((II)\) \(\frac{9}{7}\)
\((C)\) Diatomic non-rigid molecules	\((III)\) \(\frac{4}{3}\)
\((D)\) Triatomic rigid molecules	\((IV)\) \(\frac{5}{3}\)

JEE Mains 2020 Medium

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