Question

The gyromagnetic ratio of an electron in an hydrogen atom, according to Bohr model is

• A. decreases with the quantum number '$n$'.
• B. independent of which orbit it is in.
• C. negative
• D. positive

Hint:

The value of the gyromagnetic ratio for an electron is a constant approximately equal to $8.8 \times 10^{10}\ \text{C/kg}$. Since it's a ratio of two things that both scale proportionally with the orbit size and speed, those variables cancel out completely.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We must determine how the gyromagnetic ratio of a revolving electron in a hydrogen atom depends on its orbit (principal quantum number $n$).

Step 2: Key Formula or Approach:
The gyromagnetic ratio is defined as the ratio of the magnetic dipole moment ($\mu_L$) to the orbital angular momentum ($L$) of the electron.

Step 3: Detailed Explanation:
For an electron revolving in a circular orbit, the magnetic moment is $\mu_L = \frac{evr}{2}$.
Its orbital angular momentum is $L = mvr$.
Taking the ratio of the two quantities:
$$\text{Gyromagnetic Ratio} = \frac{\mu_L}{L} = \frac{\frac{evr}{2}}{mvr} = \frac{e}{2m}$$
Here, '$e$' is the elementary charge of the electron and '$m$' is the mass of the electron. Both are fundamental constants.
Because the expression $\frac{e}{2m}$ contains no variables related to the orbit (like radius $r$, velocity $v$, or quantum number $n$), the gyromagnetic ratio is a universal constant for the electron.
Therefore, it is completely independent of the orbit the electron is in.

Step 4: Final Answer:
The ratio is independent of which orbit it is in, matching option (B).