Given central force

The electron is attracted by a force $F = \frac{k}{r}$ directed towards the origin.

Use circular motion condition

For an electron moving in a circular orbit of radius $r$ with speed $v$, $\frac{mv^2}{r} = \frac{k}{r}$

Hence, $mv^2 = k$

So, $v^2 = \frac{k}{m}$

This shows that the speed $v$ is independent of $r$.

Find kinetic energy

The kinetic energy is $T = \frac{1}{2}mv^2 = \frac{1}{2}m\left(\frac{k}{m}\right) = \frac{k}{2}$

Therefore, $T_n = \frac{k}{2}$ which is independent of $n$.

Apply Bohr quantization

According to Bohr's quantization condition, $mvr = n\hbar$

Since $v = \sqrt{\frac{k}{m}}$ is constant, $r_n = \frac{n\hbar}{mv}$

Substituting $v = \sqrt{\frac{k}{m}}$, $r_n = \frac{n\hbar}{m\sqrt{k/m}} = \frac{n\hbar}{\sqrt{mk}}$

Hence, $r_n \propto n$

Compare with options

• A: $T_n \propto \frac{1}{n^2},\ r_n \propto n^2$ ❌
• B: $T_n$ independent of $n,\ r_n \propto n$ ✅
• C: $T_n \propto \frac{1}{n},\ r_n \propto n$ ❌
• D: $T_n \propto \frac{1}{n},\ r_n \propto n^2$ ❌

Final answer

The correct option is B.