Question 306: Atoms & Nuclei

Question

Using Bohr’s model, calculate the ratio of the magnetic fields generated due to the motion of the electrons in the 2
nd
and 4
th
orbits of hydrogen atom ________.

Accepted Answer

For an electron moving in a circular orbit, the magnetic field at the centre of the orbit is
$$ B=\frac{\mu_0 I}{2r} $$
where $I$ is the current due to revolving electron.
Since current is
$$ I=\frac{e}{T} $$
and the time period
$$ T=\frac{2\pi r}{v} $$
so,
$$ I=\frac{ev}{2\pi r} $$
Substituting in the formula of magnetic field,
$$ B=\frac{\mu_0}{2r}\cdot \frac{ev}{2\pi r} =\frac{\mu_0 ev}{4\pi r^2} $$
So,
$$ B \propto \frac{v}{r^2} $$
Now from BohrÃ¢ÂÂs model for hydrogen atom:
$$ r_n \propto n^2 \quad 	ext{and} \quad v_n \propto \frac{1}{n} $$
Therefore,
$$ B_n \propto \frac{1/n}{(n^2)^2} =\frac{1}{n^5} $$
Hence,
$$ \frac{B_2}{B_4}=\frac{1/2^5}{1/4^5}=\frac{4^5}{2^5}=\left(\frac{4}{2}ight)^5=2^5=32 $$
So, the required ratio is
$$ B_2 : B_4 = 32 : 1 $$

Atoms & Nuclei

Using Bohr’s model, calculate the ratio of the magnetic fields generated due to the motion of the electrons in the 2 nd and 4 th orbits of hydrogen atom ________.

Integer / Numeric Answer Type