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Step-by-Step Solution
Step 1: Write the expression for the magnetic field at the nucleus
According to Bohr’s theory, the magnetic field at the center (nucleus) of a hydrogen atom due to the motion of an electron in the nth orbit is given by:
$B = \frac{\mu_0 \, I}{2\,r_n}$
Step 2: Express the radius of the nᵗʰ orbit
From Bohr’s model, the radius of the nth orbit is:
$r_n = \frac{n^2 h^2 \varepsilon_0}{m \pi Z e^2} \implies r_n \propto n^2.$
Step 3: Express the velocity of the electron in the nᵗʰ orbit
The velocity of the electron in the nth orbit is:
$\upsilon_n = \left(\frac{e^2\,h}{2\,\varepsilon_0}\right)\frac{Z}{n} \implies \upsilon_n \propto \frac{1}{n}.$
Step 4: Relate the current to the electron’s motion
Current $I$ is given by charge per unit time. For an electron revolving around the nucleus,
$I = \frac{q}{t} = \frac{e}{\frac{2 \pi r_n}{\upsilon_n}} = \frac{e\,\upsilon_n}{2\pi\,r_n}.$
Step 5: Substitute $I$ into the expression for $B$
Substituting $I = \frac{e\,\upsilon_n}{2\pi\,r_n}$ into $B = \frac{\mu_0 \, I}{2\,r_n}$, we get:
$B = \frac{\mu_0 \left(\frac{e\,\upsilon_n}{2\pi\,r_n}\right)}{2\,r_n}
= \frac{\mu_0\, e\, \upsilon_n}{4 \pi\, r_n^2}.$
Step 6: Observe the dependence on n
Using $r_n \propto n^2$ and $\upsilon_n \propto \frac{1}{n}$, we find:
$B \propto \frac{\upsilon_n}{r_n^2}
\propto \frac{\frac{1}{n}}{(n^2)^2}
= \frac{1}{n \cdot n^4}
= \frac{1}{n^5}.$
Therefore, the magnetic field at the nucleus is proportional to $n^{-5}$.
