Magnetic Dipole due to an electron's orbital motion

In summary: Lz. Additionally, the angle θ is always positive for positive values of Lz and negative for negative values of Lz. Therefore, the only possible combination that satisfies these conditions is 2hbar, 90 for Lz = 2hbar and -2hbar, 35 for Lz = -2hbar.In summary, the correct combinations of Lz and θ for hydrogen atoms in a d state are 2hbar, 90 and -2hbar, 35. This can be determined by understanding the relationship between the magnetic dipole moment, angular momentum, and the allowed values of m for a d state.
  • #1
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Homework Statement


Select all of the following which are possible combinations of Lz and θ for hydrogen atoms in a d state, where Lz is the z component of the angular momentum L, and θ is the angle between the +zaxis and the magnetic dipole moment µℓ due to the electron's orbital motion.

-2hbar, 35
hbar, 114
2hbar, 90
2hbar, 35
2hbar, 145

Homework Equations



cosΘ= Lz/L = m(hbar)/√(l(l+1))
mu = -e/2m * L

The Attempt at a Solution



I was able to figure out that cos^-1(Lz/L) will give me all the possible combinations of the d state and the angles associated with each, but cannot figure out the angle between +z and the magnetic dipole moment. Originally I selected 2hbar, 35 as the answer but I was incorrect. I did this because it was the only answer that gave me an angle in the positive z axis and looked like the right choice through elimination. Does anyone have any ideas?
 
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  • #2


it is important to approach problems with a clear understanding of the relevant equations and concepts. In this case, the key equation is μ = -e/2m * L, where μ is the magnetic dipole moment, e is the charge of the electron, and m is the mass of the electron. This equation shows that the magnetic dipole moment is directly proportional to the angular momentum L.

Additionally, the value of Lz is related to the magnetic quantum number m, which can take on values from -l to l, where l is the orbital quantum number. For a d state, l = 2, so m can take on values of -2, -1, 0, 1, or 2.

Using these relationships, we can determine the possible combinations of Lz and θ for hydrogen atoms in a d state. Here are the steps to follow:

1. Determine the possible values of Lz based on the allowed values of m. For a d state, m can take on values of -2, -1, 0, 1, or 2. This corresponds to Lz = -2hbar, -hbar, 0, hbar, or 2hbar, respectively.

2. Use the equation cosΘ = Lz/L = m(hbar)/√(l(l+1)) to determine the angle θ for each value of Lz. For a d state, l = 2, so the equation becomes cosΘ = Lz/√(2(2+1)) = Lz/√6. Plugging in the values of Lz from step 1, we get θ = cos^-1(-2/√6), cos^-1(-1/√6), cos^-1(0), cos^-1(1/√6), and cos^-1(2/√6) for Lz = -2hbar, -hbar, 0, hbar, and 2hbar, respectively.

3. Look for the given combinations of Lz and θ in the list of possible values calculated in step 2. The correct answers are 2hbar, 90 and -2hbar, 35. This is because the magnetic dipole moment μ is always perpendicular to the z axis, so the angle θ must be 90 degrees for any non-zero
 

Related to Magnetic Dipole due to an electron's orbital motion

1. What is a magnetic dipole?

A magnetic dipole is a type of magnet that has two poles, a north pole and a south pole, separated by a small distance. This type of magnet can be created by the movement of electric charges, such as electrons, and is responsible for the magnetic field around a magnet.

2. How is a magnetic dipole created by an electron's orbital motion?

As an electron moves in an orbit around an atomic nucleus, it creates a small electric current. This current, together with the electron's spin, creates a magnetic field around the electron, making it a magnetic dipole.

3. Does the magnetic dipole of an electron affect its orbit?

Yes, the magnetic dipole of an electron can affect its orbit. This is because the magnetic field created by the dipole can interact with other magnetic fields, such as those of other electrons or external magnetic fields, and change the direction or shape of the electron's orbit.

4. How does the strength of an electron's magnetic dipole compare to other types of magnets?

The magnetic dipole of an electron is relatively weak compared to other types of magnets, such as permanent magnets. This is because an electron's dipole moment is very small and its magnetic field is only present when the electron is in motion.

5. Can the magnetic dipole of an electron be measured?

Yes, the magnetic dipole of an electron can be measured using various techniques, such as electron spin resonance or nuclear magnetic resonance. These techniques use the interaction between the electron's magnetic field and an external magnetic field to measure the strength of the dipole moment.

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