Question

The relation between magnetic moment 'M' of revolving electron and principle quantum number 'n' is

• A. $M \propto \frac{1}{n}$
• B. $M \propto n$
• C. $M \propto n^2$
• D. $M \propto n^3$

Hint:

The magnetic moment of an electron in the $n$-th orbit is exactly $n$ times the Bohr magneton ($M = n\mu_B$). This makes it linearly proportional to the principal quantum number.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to determine the mathematical proportionality between the magnetic dipole moment ($M$) of an electron revolving in a Bohr orbit and its principal quantum number ($n$).

Step 2: Key Formula or Approach:
The magnetic moment of a revolving charge is given by $M = I \times A$.
A more direct relationship for an electron is through the gyromagnetic ratio: $M = \frac{e}{2m_e} L$, where $L$ is the orbital angular momentum.
According to Bohr's second postulate of quantization, angular momentum is $L = \frac{nh}{2\pi}$.

Step 3: Detailed Explanation:
Start with the relation between magnetic moment and angular momentum:
$$M = \frac{e}{2m_e} L$$ Substitute Bohr's quantization condition for $L$:
$$M = \frac{e}{2m_e} \left( \frac{nh}{2\pi} \right)$$ Combine the terms to isolate $n$:
$$M = \left( \frac{eh}{4\pi m_e} \right) n$$ The quantity in the parentheses ($\frac{eh}{4\pi m_e}$) is known as the Bohr magneton ($\mu_B$), which is a fundamental constant for an electron.
Since $\mu_B$ is constant, we can clearly see the direct proportionality:
$$M \propto n$$

Step 4: Final Answer:
The correct relation is $M \propto n$, matching option (B).