Q. Suppose an electron is attracted towards the origin by a force $\frac{k}{r}$ 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
A) $T_{n}$ independent of $n,\ r_{n} \propto n$
B) $T_{n} \propto \frac{1}{n},\ r_{n} \propto n$
C) $T_{n} \propto \frac{1}{n},r_{n} \propto n^{2}$
D) $T_{n} \propto \frac{1}{n^{2}},r_{n} \propto n^{2}$
Explanation
(a) According to Bohr theory, $mvr = n\frac{h}{2\pi} \Rightarrow v = \frac{nh}{2\pi mr}$
and $\frac{mv^{2}}{r} \propto \frac{k}{r} \Rightarrow \frac{m}{r}\left( \frac{n^{2}h^{2}}{4\pi^{2}m^{2}r^{2}} \right) \propto \frac{k}{r} \Rightarrow r_{n} \propto n$
Kinetic energy $T = \frac{1}{2}mv^{2} = \frac{1}{2}m\left( \frac{n^{2}h^{2}}{4\pi^{2}m^{2}r^{2}} \right) \Rightarrow T_{n} \propto \frac{n^{2}}{r^{2}}$
But as $r \propto n$ therefore $T \propto n^{0}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.