Step 1: Understand the question.
We must find how the gyromagnetic ratio of an electron in a hydrogen atom depends on the orbit, using the Bohr model.
Step 2: Define the gyromagnetic ratio.
The gyromagnetic ratio is the magnetic moment of the orbiting electron divided by its angular momentum: ratio $= \dfrac{\mu_L}{L}$.
Step 3: Write the magnetic moment.
For an electron going around a circle of radius $r$ with speed $v$, the magnetic moment is:
\[ \mu_L = \frac{evr}{2} \]
Step 4: Write the angular momentum.
The orbital angular momentum is:
\[ L = mvr \]
Step 5: Take the ratio.
Divide the two. The $v$ and $r$ cancel out:
\[ \frac{\mu_L}{L} = \frac{evr/2}{mvr} = \frac{e}{2m} \]
Step 6: Read the result.
The answer $\dfrac{e}{2m}$ has only the charge and mass of the electron, both fixed constants. It has no orbit number $n$, no radius, and no speed. So the ratio is the same for every orbit. It is independent of which orbit the electron is in. This matches option (2).
\[ \boxed{\frac{\mu_L}{L} = \frac{e}{2m}\ \text{(same for all orbits)}} \]