Question

In Bohr's theory of hydrogen atom, 'r' is the radius of the orbit, 'V' is the speed of electron and E is the total energy of the electron. Out of the following which physical quantity is inversely proportional to principal quantum number?

• A. \(\frac{1}{Vr}\)
• B. \(Vr\)
• C. \(V^2r\)
• D. \(Vr^2\)

Hint:

Use $r\propto n^2$ and $V\propto 1/n$, then combine them for each option.

Answer 1

The Correct Option is A

Step 1: Use angular momentum quantisation
Bohr gave $mVr=\frac{nh}{2\pi}$, so $Vr=\frac{nh}{2\pi m}$.
Since $h$, $m$ and $\pi$ are constants, $Vr$ is directly proportional to $n$.

Step 2: Pick the reciprocal
If $Vr\propto n$ then $\frac{1}{Vr}\propto\frac{1}{n}$, which is the inverse dependence the question asks for.
This matches option (A).

Step 3: Check the rest quickly
Using $r\propto n^2$ and $V\propto 1/n$: $V^2r$ stays constant, $Vr^2\propto n^3$, and $Vr\propto n$.
None of these falls as $n$ rises, so (B), (C) and (D) are out.

Final Answer:
Since $Vr\propto n$, the quantity $\frac{1}{Vr}$ falls as $1/n$. Option (A). \[ \boxed{\text{(A)}} \]