The magnetic field produced at the centre of the nucleus by the electron revolving in the 1^ st orbit of hydrogen atom is B₁ and that by the electron revolving in the 2^ nd orbit of He ⁺ ion is B₂ . If the ratio B₁ B₂ is x , then the value of x is _____.

Step-by-step solution

The magnetic field at the centre of a circular orbit is given by B = ₀ i 2r . The equivalent current is i = e T = ev 2 r . Thus, B = ₀ e v 4 r^2 . From Bohr's model, velocity v Z n and radius r n^2 Z . Substituting these dependencies, we get: B Z/n (n^2/Z)^2 B Z^3 n^5 . For the 1^ st orbit of hydrogen atom ( Z=1, n=1 ): B₁ 1^3 1^5 = 1 . For the 2^ nd orbit of He ⁺ ion ( Z=2, n=2 ): B₂ 2^3 2^5 = 8 32 = 1 4 . The ratio is B₁ B₂ = 1 1/4 = 4 . Hence, the value of x is 4 . Answer: 4