What is the energy difference between parallel and antiparallel alignment of the z component of an electron’s spin magnetic dipole moment with an external magnetic field of magnitude 0.25 T, directed parallel to the z axis?
The correct answer and explanation is:
The energy difference between the parallel and antiparallel alignment of an electron’s spin magnetic dipole moment with an external magnetic field is given by the equation:
[
\Delta E = 2 \mu_B B
]
where:
- ( \mu_B ) is the Bohr magneton, given by: [
\mu_B = \frac{e \hbar}{2m_e} \approx 9.274 \times 10^{-24} \text{ J/T}
] - ( B ) is the external magnetic field strength (in this case, 0.25 T).
Calculation:
[
\Delta E = 2 (9.274 \times 10^{-24} \text{ J/T}) (0.25 \text{ T})
]
[
\Delta E = 4.637 \times 10^{-24} \text{ J}
]
or in electron volts (( 1 \text{ eV} = 1.602 \times 10^{-19} ) J):
[
\Delta E = \frac{4.637 \times 10^{-24} \text{ J}}{1.602 \times 10^{-19} \text{ J/eV}}
]
[
\Delta E \approx 2.90 \times 10^{-5} \text{ eV}
]
Explanation:
The energy difference arises due to the interaction between the electron’s intrinsic magnetic moment and the applied magnetic field. Electrons have an intrinsic spin magnetic dipole moment, which can align parallel or antiparallel to an external magnetic field.
- When the electron’s spin is parallel to the field, it is in a lower energy state.
- When the spin is antiparallel, it is in a higher energy state.
- The energy difference ( \Delta E ) between these two states determines the splitting of energy levels, which is crucial in electron spin resonance (ESR) and Zeeman splitting observed in spectroscopy.
This energy splitting is essential in quantum mechanics and condensed matter physics, influencing atomic transitions, magnetic resonance imaging (MRI), and quantum computing.
Now, I’ll generate an image representing the energy splitting of electron spin in a magnetic field.
Here is the generated scientific diagram illustrating the Zeeman effect for an electron’s spin in a magnetic field. It visually represents the energy splitting between the parallel and antiparallel spin states due to the external magnetic field. Let me know if you need any modifications or further explanations!
