An electron moves around the nucleus in a hydrogen atom of radius 0.05 nm with a velocity of \( 2\times10^{6}\,\mathrm{m\,s^{-1}} \). The magnetic field produced at the centre of nucleus is :
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 12.8 T
An electron going round in a circle repeats its path many times a second, so it acts as a current loop. The current is the charge passing a point per second, and the field at the centre of the loop follows from the Biot-Savart law.
\( \displaystyle I = \frac{e}{T} = \frac{e\,v}{2\pi r} \)
\( \displaystyle B = \frac{\mu_0 I}{2r} = \frac{\mu_0\,e\,v}{4\pi r^{2}} \)
\( r = 0.05\,\mathrm{nm} = 5\times10^{-11}\,\mathrm{m} \)
\( r^{2} = 2.5\times10^{-21}\,\mathrm{m^2} \)
\( \mu_0/4\pi = 10^{-7}\,\mathrm{T\,m\,A^{-1}} \)
\( \displaystyle B = \frac{10^{-7} \times 1.6\times10^{-19} \times 2\times10^{6}}{2.5\times10^{-21}} \)
\( \displaystyle B = \frac{3.2\times10^{-20}}{2.5\times10^{-21}} = 12.8\,\mathrm{T} \)