An electron revolves anti-clockwise around a proton in a hydrogen atom. The speed of electron is v and radius of its circular orbit is r. find
(i) The magnetic field produced at the centre
(ii) Magnetic dipole moment of circulating electron
(iii) The ratio of magnetic moment of angular momentum of electron
Strategy: A charge in motion constitutes current. Therefore, an electron moving on circle is equivalent to a current carrying loop

Explanation

The equivalent current is `i = (e)/(T)`
or ` i= (ev)/(2pi r) " " ( :' T = (2pi r)/(v))`
(i) Magnetic field at the centre is `B = (mu_(0)i)/(2r)`
`:. B = (mu_(0) eV)/(4pi r^(2)) ox`
(ii) Magnetic dipole moment is `M = ixx A`
`:. M = (ev)/(2pi r) xx pi r^(2) or M = (evr)/(2)`
(iii) `("Magnetic moment")/("Angular momentum") = (evr//2)/(mvr) = (e)/(2m)`. (This is a constant called gyromagnetic ratio)