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