Question

Using Bohr model, calculate the electric current created by the electron when the H-atom is in the ground state.

Answer 1

The equivalent electric current due to rotation of charge is given by `i = Q/T = Q(1/T) = Q xx f`, where `f` is the frequency.

In a Hydrogen atom, an electron in the ground state revolves in a circular orbit whose radius is equal to the Bohr radius (a₀). Let the velocity of the electron is v.

∴ Number of revolutions per unit time `f = (2πa_0)/v`

The electric current is given by i = Q of, if Q charge flows in time T. Here, Q = e.

The electric current is given by `i = e((2πa_0)/v) = (2πa_0)/v e`.

Important points:

Some other quantities for a revolution of electron in nth orbit
	Quantity	Formula	Dependency on n and Z
1.	Angular speed	`ω_n = v_n/r_n = (pimZ^2e^4)/(2ε_0^2n^3h^3)`	`ω_n ∝ Z^2/n^3`
2.	Frequency	`v_n = ω_n/(2pi) = (mZ^2e^4)/(4ε_0^2n^3h^3)`	`v_n ∝ Z^2/n^3`
3.	Time period	`T_n = 1/v_n = (4ε_0^2n^3h^3)/(mZ^2e^4)`	`T_n ∝ n^3/Z^2`
4.	Angular momentum	`L_n = mv_nr_n = n(h/(2pi))`	`L_n ∝ n`
5.	Corresponding current	`i_n = ev_n = (mZ^2e^5)/(4ε_0^2n^3h^3)`	`i_n ∝ Z^2/n^3`
6.	Magnetic moment	`M_n = i_nA = i_n (pir_n^2)` (where `mu_0 = (eh)/(4pim)` Bhor magneton)	`M_n ∝ n`
7.	Magnetic field	`B = (mu_0i_n)/(2r_n) = (pim^2Z^3e^7 m_0)/(8ε_0^3n^5h^5)`	`B ∝ Z^3/n^5`