Question

The force acting on the electron in hydrogen atom (Bohr's theory) is related to the principal quantum number n as

• A. \(n^4\)
• B. \(n^{-4}\)
• C. \(n^2\)
• D. \(n^{-2}\)

Hint:

Force is k e^2 / r^2 and r is proportional to n squared.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
In Bohr's theory the electrostatic force on the electron is \(F = \dfrac{ke^2}{r^2}\), and the radius of the \(n\)-th orbit is \(r_n = n^2r_1\).

Step 2: Substitute:
\[ F\propto\frac1{r^2} = \frac1{(n^2)^2} = n^{-4} \]

Step 3: Check:
Option (B). The speed goes as \(1/n\) and the radius as \(n^2\), so \(\dfrac{mv^2}r\propto\dfrac{n^{-2}}{n^2} = n^{-4}\), which agrees.

Final Answer:
r goes as n squared, so F goes as n to the minus 4. \[ \boxed{\text{(B) }n^{-4}} \]