Question:
Magnetic field at the centre of the hydrogen atom due to motion of electron in nth orbit is proportional to
Magnetic field at the centre of the hydrogen atom due to motion of electron in nth orbit is proportional to
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Physics Tip : As the principal quantum number $n$ increases, both the current decreases and the radius increases, leading to a very rapid decrease ($1/n^5$) in the central magnetic field.
Updated On: Apr 23, 2026
- $n^{4}$
- $n^{-3}$
- $n^{3}$
- $n^{-5}$
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The Correct Option is D
Solution and Explanation
Concept:
Physics (Atoms) – Magnetic Field produced by Bohr's Orbiting Electron.
Step 1: Identify proportionalities for radius and angular velocity. In Bohr's model for the $n^{th}$ orbit:
• Radius $r_n \propto n^2$
• Angular velocity $\omega_n \propto \frac{1}{n^3}$
Step 2: Relate current to angular velocity. The equivalent current $I_n$ due to the orbiting electron is $I_n = \frac{q\omega_n}{2\pi}$. Therefore, $I_n \propto \omega_n \propto \frac{1}{n^3}$.
Step 3: Determine the magnetic field proportionality. The magnetic field at the center of a circular current loop is $B_n = \frac{\mu_0 I_n}{2r_n}$. $$B_n \propto \frac{I_n}{r_n} \propto \frac{1/n^3}{n^2}$$ $$B_n \propto \frac{1}{n^5} \text{ or } n^{-5}$$ $$ \therefore \text{The magnetic field } B_{n} \text{ is proportional to } n^{-5}. $$
Step 1: Identify proportionalities for radius and angular velocity. In Bohr's model for the $n^{th}$ orbit:
• Radius $r_n \propto n^2$
• Angular velocity $\omega_n \propto \frac{1}{n^3}$
Step 2: Relate current to angular velocity. The equivalent current $I_n$ due to the orbiting electron is $I_n = \frac{q\omega_n}{2\pi}$. Therefore, $I_n \propto \omega_n \propto \frac{1}{n^3}$.
Step 3: Determine the magnetic field proportionality. The magnetic field at the center of a circular current loop is $B_n = \frac{\mu_0 I_n}{2r_n}$. $$B_n \propto \frac{I_n}{r_n} \propto \frac{1/n^3}{n^2}$$ $$B_n \propto \frac{1}{n^5} \text{ or } n^{-5}$$ $$ \therefore \text{The magnetic field } B_{n} \text{ is proportional to } n^{-5}. $$
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