Suppose an electron is attracted towards the origin by a force $\mathrm{k} / \mathrm{r}$ where ' k ' is a constant and ' r ' is the distance of the electron from the origin. By applying Bohr model to this system, the radius of the $\mathrm{n}^{\text {th }}$ orbital of the electron is found to be ' $\mathrm{r}_\mathrm{n}$ ' and the kinetic energy of the electron to be $\mathrm{T}_\mathrm{n}$. Then which of the following is true?

(B) $T_{n}$ independent of $n, r_{n} \propto n$

(A) $T_{n} \propto 1 / n^{2}, \quad r_{n} \propto n^{2}$

(C) $T_{n} \propto 1 / n, \quad r_{n} \propto n$

(D) $T_{n} \propto 1 / n, \quad r_{n} \propto n^{2}$

JEE Main · 2008 · 27 Apr 2008 · Offline · Modern Physics

Suppose an electron is attracted towards the origin by a force

k / r

where ' k ' is a constant and ' r ' is the distance of the electron from the origin. By applying Bohr model to this system, the radius of the

n^{th}

orbital of the electron is found to be '

r_{n}

' and the kinetic energy of the electron to be

T_{n}

. Then which of the following is true?

Answer: (B)

If the attractive force is

F = \frac{k}{r}

, equating it to the centripetal requirement

\frac{m v ^{2}}{r} = \frac{k}{r}

shows

m v^{2} = k

, a constant independent of the orbit. The kinetic energy

T = \frac{1}{2} m v^{2} = \frac{k}{2}

is then the same for every orbit and does not depend on

n

. Bohr's quantisation

m v r = n ℏ

fixes the radius: since

m v

is constant,

r

is proportional to

n

. So the kinetic energy is independent of

n

while the orbital radius grows in direct proportion to

n

JEE Main · 2008 · 27 Apr 2008 · Offline · Single correct option