In the hydrogen atom, an electron revolves around the nucleus in a circular orbit of radius $5.11 × 10^{-11} m$ with a frequency of $6.8 × 10^{15} Hz$. The magnetic field at the centre of the orbit is:

13.4 T

The correct answer is Option (2) → 13.4 T

Given:

Radius of orbit: $r = 5.11 \times 10^{-11} \ \text{m}$

Frequency: $f = 6.8 \times 10^{15} \ \text{Hz}$

Charge of electron: $e = 1.6 \times 10^{-19} \ \text{C}$

Permeability: $\mu_0 = 4 \pi \times 10^{-7} \ \text{Tm/A}$

Current due to orbital motion:

$I = e f = (1.6 \times 10^{-19})(6.8 \times 10^{15})$

$I = 1.088 \times 10^{-3} \ \text{A}$

Magnetic field at the centre of a current loop:

$B = \frac{\mu_0 I}{2r}$

Substitute values:

$B = \frac{(4 \pi \times 10^{-7})(1.088 \times 10^{-3})}{2(5.11 \times 10^{-11})}$

$B = \frac{1.365 \times 10^{-9}}{1.022 \times 10^{-10}}$

$B \approx 13.4 \ \text{T}$

Final Answer: $B \approx 13.4 \ \text{T}$