Which among the following expressions is the correct for the magnetic field at the site of the nucleus in a hydrogen atom due to the circular motion of the electron? Assume that the atom is in its ground state and the expressions are given in terms of the fundamental constants. where m = mass of an electron, e = charge of an electron, h = planck's constant)

Step-by-step solution

We have mv 2 r = e 2 4 π ε 0 r 2 or,  v 2 r = e 2 4 π ε 0 m ...(i) From Bohr's quantization rule, in ground state, vr = h 2 π m · ...(ii) From (i) and (ii), v = e 2 2 ε 0 h ...(iii) and r = ε 0 h 2 π me 2 ...(iv) As the electron moves along a circle, it crosses any point on the circle v 2 π r times per unit time. The charge crossing per unit time, that is the current, is i = ev 2 π r . The magnetic field at the centre due to this circular current is B