The electric potential between a proton and an electron is given by V = V 0 ln r r 0 where r 0 is a constant. Assuming Bohr’s model to be applicable, write variation of r n with n , n being the principal quantum number.

U = eV = eV 0 ln ( r r 0 ) | F | = | - d U d r | = e V 0 r This force provides necessary centripetal force. Hence, m v 2 r = e V 0 r Or v = e V 0 m ... (i) Moreover, m v r = n h 2 π ... (ii) Dividing Eq. (ii) by (i), we have mr = nh 2 π m eV 0 or r n ∝ n

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PhysicsModern PhysicsAtomic structureEasy2 minPYQ_2020

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The electric potential between a proton and an electron is given by $V = V_{0} ln \frac{r}{r_{0}}$ where r₀ is a constant. Assuming Bohr’s model to be applicable, write variation of $r_{n}$ with $n$ , $n$ being the principal quantum number.

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Solution:

U = eV = eV₀ln $(\frac{r}{r_{0}})$

$| F | = | - \frac{d U}{d r} | = \frac{e V_{0}}{r}$

This force provides necessary centripetal force.

Hence, $\frac{m v^{2}}{r} = \frac{e V_{0}}{r}$

Or $v = \sqrt{\frac{e V_{0}}{m}}$ ... (i)

Moreover, $m v r = \frac{n h}{2 π}$ ... (ii)

Dividing Eq. (ii) by (i), we have

$mr = (\frac{nh}{2 π}) \sqrt{\frac{m}{{eV}_{0}}}$ or $r_{n} \propto n$

Stream:NTA_ABHYAS_NEETSubject:PhysicsTopic:Modern PhysicsSubtopic:Atomic structure

⏱ 2mℹ️ Source: PYQ_2020

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