If an electron is moving in the $\mathrm{n}^{\text {th }}$ orbit of the hydrogen atom, then its velocity $\left(v_{\mathrm{n}}\right)$ for the $\mathrm{n}^{\text {th }}$ orbit is given as :

(A) $v_{n} \alpha \frac{1}{n}$

(D) $v_{n} \alpha \frac{1}{n^{2}}$

JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Modern Physics

If an electron is moving in the

n^{th}

orbit of the hydrogen atom, then its velocity

(v_{n})

for the

n^{th}

orbit is given as :

(A) $v_{n} α \frac{1}{n}$ ✓ Correct
(B) $v_{n} α n^{2}$
(C) $v_{n} α n$
(D) $v_{n} α \frac{1}{n ^{2}}$

Answer: (A)

Solution

Bohr's quantisation condition gives

m v_{n} r_{n} = \frac{nh}{2 π}

, so

v_{n} \propto \frac{n}{r _{n}}

. The orbit radius grows as

r_{n} \propto n^{2}

, hence

v_{n} \propto \frac{n}{n ^{2}} = \frac{1}{n}

The orbital speed of the electron varies as

v_{n} \propto \frac{1}{n}

JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Single correct option

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