Physics 309
Magnetic Interactions in the Hydrogen Atom


  1. In the Bohr model of the hydrogen atom (i.e. much like the classical mini-solar system picture), the electron is in a circular orbit at a radius of $r=0.53~{\rm\AA}$. What is the current produced by the electron in terms of its speed $v$, its charge $e$, and the radius of its orbit $r$? Show that

    \begin{displaymath} 2 \pi r I = e v \qquad . \end{displaymath}

    where $I$ is the current produced by the electron.


  2. For the electron in the Bohr atom use the Biot-Savart Law to show that the magnetic field produced at the center of the orbit (i.e., at the location of the proton) is

    \begin{displaymath} \vec B = {\mu_0 \over 4 \pi} {e \vec v \times \vec r \over r^3} \end{displaymath}

    where $\vec v$ is the electron velocity and $\vec r$ is the electron position vector. The expression above uses MKS units.


  3. What is the electric field produced by the electron at the position of the proton? Use this expression to show that

    \begin{displaymath} {\vec r \over r^3} = {4 \pi \epsilon_0 \over e} \vec E \end{displaymath}

    where $\vec E$ is the electron's electric field. The expression above uses MKS units.


  4. Use the above results to show that the magnetic field produced by the electron at the position of the proton is

    \begin{displaymath} \vec B = \epsilon_0 \mu_0 ~ \vec v \times \vec E \qquad . \end{displaymath}