Magnetic Field Seen by Orbiting Electron

In summary, the magnetic field seen by an electron in the first Bohr orbit of hydrogen due to the intrinsic spin moment of the proton is given by the equation \mathbf B = \frac{Ze^2}{emc^{2}r^2}\frac{\mathbf{L}}{4 \pi \epsilon_{0}}. This is calculated by combining Ampere's law, Coulomb's law, and the spin-orbit interaction energy.
  • #1
atomicpedals
209
7

Homework Statement



What is the magnetic field seen by an electron in the first Bohr orbit of hydrogen due to the intrinsic spin moment of the proton?

2. The attempt at a solution

I think I've got it; but might be way off.

A charged nucleus with velocity -v constitutes a current element j,
[tex]\mathbf j = -Ze \mathbf v [/tex]
Ampere's law then gives
[tex]\mathbf B = - \frac{Ze\mu}{4\pi} \frac{\mathbf v \times \mathbf r}{r^{3}}[/tex]
Coulomb's law
[tex]\mathbf E = - \frac{Ze}{4\pi \epsilon}\frac{\mathbf r}{r^{3}}[/tex]
and thus
[tex]\mathbf B = - \frac{1}{c^{2}}\mathbf v \times \mathbf E[/tex]
The spin-orbit interaction energy is found to be
[tex]\Delta E = \frac{1}{2} \frac{g \mu}{\hbar} \mathbf S \cdot \mathbf B[/tex]
Applying the electric field E and the force F acting on the electron
[tex]-e \mathbf E = \mathbf F[/tex]
[tex]\mathbf F = - \frac{dV(r)}{dr}\frac{\mathbf r}{r}[/tex]
and so finally
[tex]\mathbf B = - \frac{1}{ec^{2}}\frac{1}{r}\frac{dV(r)}{dr}\mathbf v \times \mathbf r[/tex]
[tex]\mathbf B = - \frac{1}{emc^{2}}\frac{1}{r}\frac{dV(r)}{dr}\mathbf{L} [/tex]
 
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  • #2
where the angular momentum is found to be\mathbf L = m \mathbf r \times \mathbf vFor hydrogen, the electric potential is given byV(r) = - \frac{Z e^{2}}{4 \pi \epsilon_{0} r}Substituting in and rearranging gives\mathbf B = \frac{Ze^2}{emc^{2}r^2}\frac{\mathbf{L}}{4 \pi \epsilon_{0}}So the resulting magnetic field seen by an electron in the first Bohr orbit due to the intrinsic spin moment of the proton is\mathbf B = \frac{Ze^2}{emc^{2}r^2}\frac{\mathbf{L}}{4 \pi \epsilon_{0}}
 

1. What is a magnetic field seen by an orbiting electron?

A magnetic field seen by an orbiting electron is a phenomenon in which an electron's orbit around a nucleus is affected by an external magnetic field. This can cause the electron's path to change, leading to changes in its energy and behavior.

2. How is a magnetic field seen by an orbiting electron measured?

A magnetic field seen by an orbiting electron can be measured using a magnetic field detector, such as a magnetometer. This device measures the strength and direction of the magnetic field, which can then be used to calculate the effect on the electron's orbit.

3. What factors affect the strength of a magnetic field seen by an orbiting electron?

The strength of a magnetic field seen by an orbiting electron depends on the strength of the external magnetic field, the distance between the electron and the nucleus, and the speed of the electron's orbit. Additionally, the type of atom and the number of electrons in the orbit can also affect the strength of the magnetic field.

4. How does a magnetic field seen by an orbiting electron impact atoms and molecules?

A magnetic field seen by an orbiting electron can impact atoms and molecules in various ways. It can alter the energy levels of electrons, leading to changes in chemical bonding and molecular properties. It can also affect the movement and behavior of charged particles within a substance.

5. Can a magnetic field seen by an orbiting electron be manipulated for practical applications?

Yes, a magnetic field seen by an orbiting electron can be manipulated for practical applications. This phenomenon is utilized in various technologies, such as magnetic resonance imaging (MRI) and particle accelerators. It is also important in understanding the behavior of materials in magnetic fields, which has applications in engineering and materials science.

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