Question

If '$\mu$' is the magnetic moment of an electron revolving around hydrogen nucleus in an orbit of principal quantum number 'n', then

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

Hint:

Magnetic Moment in Bohr Orbits

• In atomic physics, magnetic moment arises from electron motion.
• $\mu = \dfrace2m_e L$ where $L = n\hbar$.
• $\mu = n \mu_B$, where $\mu_B = \dfrace\hbar2m_e$.
• This leads to $\mu \propto n$ for Bohr model electrons.

Answer 1

The Correct Option is B

The magnetic moment $\mu$ of a revolving electron is related to its angular momentum $L$ by the expression: \[ \mu = \frac{e}{2m_e} L \] In Bohr’s atomic model, the angular momentum of the electron in the $n^\text{th}$ orbit is quantized as: \[ L = n\hbar \] Substituting into the magnetic moment equation: \[ \mu = \frac{e}{2m_e} \cdot n\hbar = n \cdot \left( \frac{e\hbar}{2m_e} \right) \] The term in parentheses is a constant called the Bohr magneton ($\mu_B$). Thus: \[ \mu = n\mu_B \Rightarrow \mu \propto n \] So, the magnetic moment is directly proportional to the principal quantum number $n$. Hence, option (2) is correct.

Answer 2

Step 1: Understand the magnetic moment of an electron in an atom
The magnetic moment (μ) of an electron revolving around the nucleus arises due to its orbital motion, which creates a current loop and hence a magnetic field.

Step 2: Relationship between magnetic moment and angular momentum
The magnetic moment is proportional to the orbital angular momentum (L) of the electron:
μ ∝ L

Step 3: Angular momentum quantization in Bohr’s model
According to Bohr’s quantization, the orbital angular momentum is given by:
L = nħ
where n is the principal quantum number and ħ is the reduced Planck constant.

Step 4: Deriving the relation of magnetic moment with principal quantum number
Since μ ∝ L and L = nħ, it follows that:
μ ∝ n
Thus, the magnetic moment increases linearly with the principal quantum number n.

Step 5: Conclusion
Therefore, the magnetic moment μ of an electron revolving in the nth orbit is directly proportional to n, i.e., μ ∝ n.