Question

In the Bohr model of hydrogen atom, the electron follows a circular orbit centred on the
nucleus containing a proton. The motion of the electron along the circular orbit
constitutes a current. Calculate the magnetic field produced by the orbiting electron at the
site of the proton.

Answer 1

Use Bohr's law

$$m _ {e} \cdot V _ {n} \cdot r _ {n} = \frac {n \cdot h}{2 \cdot \pi}$$ $$m _ {e} \cdot \omega_ {n} \cdot r _ {n} ^ {2} = \frac {n \cdot h}{2 \cdot \pi}$$

Acceleration of electron

$$a _ {n} = \frac {V _ {n} ^ {2}}{r _ {n}} = \omega_ {n} ^ {2} \cdot r _ {n}$$ $$\frac {q _ {e} ^ {2}}{4 \pi \cdot \varepsilon_ {0} \cdot r _ {n} ^ {2} \cdot m _ {e}} = \omega_ {n} ^ {2} \cdot r _ {n}$$

Solve this equations

$$\omega_ {n} = \frac {1}{2 \cdot n ^ {3} \cdot h ^ {3}} \cdot \frac {m _ {e}}{\varepsilon_ {0} ^ {2}} \cdot q _ {e} ^ {4} \cdot \pi$$ $$r _ {n} = n ^ {2} \cdot h ^ {2} \cdot \frac {\varepsilon_ {0}}{q _ {e} ^ {2} \cdot m _ {e} \cdot \pi}$$

The electron current is

$$I = q _ {e} \cdot \frac {\omega_ {n}}{2 \cdot \pi}$$ $$I = \frac {1}{4} \cdot \frac {q _ {e} ^ {5}}{n ^ {3} \cdot h ^ {3}} \cdot \frac {m _ {e}}{\varepsilon_ {0} ^ {2}}$$

Use Biot-Savart law:

$$B = \mu_ {0} \frac {I}{2 r _ {n}}$$ $$B = \frac {1}{8} \mu_ {0} \frac {q _ {e} ^ {7}}{n ^ {5} h ^ {5}} \frac {m _ {e} ^ {2}}{\varepsilon_ {0} ^ {3}} \pi$$

Answer is 12.5 T

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