NEET-XII-Physics
43: Bohr's Model and Physics of the Atom
- #26Show that the ratio of the magnetic dipole moment to the angular momentum (l = mvr) is a universal constant for hydrogen-like atoms and ions. Find its value.Ans : Mass of the electron, m = 9.1×10`` -``31kg
Radius of the ground state, r = 0.53×10`` -``10m
Let f be the frequency of revolution of the electron moving in the ground state and A be the area of orbit.
Dipole moment of the hydrogen like elements (μ) is given by
μ = niA = qfA
`` =e\times \frac{m{e}^{4}}{4{\in }_{0}^{2}{h}^{3}{n}^{3}}\times \left(\,\mathrm{\,\pi \,}{r}_{0}^{2}{n}^{2}\right)``
`` =\frac{m{e}^{5}\times \left(\,\mathrm{\,\pi \,}{{r}_{0}}^{2}{n}^{2}\right)}{4{\in }_{0}^{2}{h}^{3}{n}^{3}}``
`` ``
Here,
h = Planck's constant
e = Charge on the electron
`` {\epsilon }_{0}`` = Permittivity of free space
n = Principal quantum number
Angular momentum of the electron in the hydrogen like atoms and ions (L) is given by
`` L=mvr=\frac{nh}{2\,\mathrm{\,\pi \,}}``
Ratio of the dipole moment and the angular momentum is given by
`` \frac{\mu }{L}=\frac{{e}^{5}\times m\times \,\mathrm{\,\pi \,}{r}^{2}{n}^{2}}{4{\in }_{0}{h}^{3}{n}^{3}}\times \frac{2\,\mathrm{\,\pi \,}}{nh}``
`` \frac{\mu }{L}=\frac{{\left(1.6\times {10}^{-19}\right)}^{5}\times \left(9.10\times {10}^{-31}\right)\times {\left(3.14\right)}^{2}\times {\left(0.53\times {10}^{-10}\right)}^{2}}{2\times {\left(8.85\times {10}^{-12}\right)}^{2}\times {\left(6.63\times {10}^{-34}\right)}^{3}\times {1}^{2}}``
`` \frac{\mu }{L}=3.73\times {10}^{10}\,\mathrm{\,C\,}/\,\mathrm{\,kg\,}``
Ratio of the magnetic dipole moment and the angular momentum do not depends on the atomic number 'Z'.
Hence, it is a universal constant.
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