Expension for
Morennent
magnetic
of revoning electron
Answers
Answer:
Let us consider an electron that is revolving around in a circle of radius r with a velocity v. The charge of the electron is e and its mass is m, both of which are constant. The time period T of the electrons’ orbit is:
T = CircumferenceVelocity = 2πrv
The current i due to the motion of the electron is the charge flowing through that time period,
i = −e2πrv = −ev2πr
Note, that the current is in the opposite direction as the electron is negatively charged. The magnetic moment due to a current loop enclosing an area A is given by:
μ = iA
Magnetic moment of an electron:
μ = −ev2πr A = −ev2πr πr2
μ = −erv2
Let us divide and multiply by the mass of the electron,
μ = −e2me me vr
We know, that the angular momentum L is given by:
L = mvr
Thus, we can write,
μ = −e2me L
Since, the angular momentum is given by the right-hand rule with respect to the velocity and the current in the opposite direction hence, the negative sign shows that the two quantities are on opposing directions as shown in the figure,
μ→ = −e2me L→
This is an important result as the magnetic moment is only dependent upon the angular momentum. This is why the orbital angular momentum and orbital magnetic moment terms are used interchangeably. The same is true for the spin angular moment.
For an electron revolving in an atom, the angular momentum is quantized as proposed by Niels Bohr.
The angular momentum is given by:
L = n h2π, n = 0, ±1, ±2 …
Where n is the orbit quantum number and h is the Planck’s constant,
μ = n −e2me h2π
μ = −n eh4πme
The quantity that multiplies with n is constant and is known as the Bohr Magneton μB,
μB = eh4πme = 9.27 × 10−27 J⁄T
Following is the table of links related to magnetic characteristic:
Magnetization and Magnetic Intensity
Magnetic Classification of Materials
The Bohr Magneton is used very widely to express magnetic moments at the atomic scale.
The expression we obtained is good for only simple atoms like hydrogen and does not predict all the magnetic moment states of an electron. As you would have learned in chemistry, the electron does not really revolve around the nucleus. Instead, the electron’s orbital magnetic moment is obtained by virtue of being trapped in the nuclei (plural of nucleus) potential well.
The spin and orbital magnetic moments of atoms combine vectorially in a sample to produce the net magnetic moment of that particular sample. It is these magnetic moments obtained by the combination of orbital and spin magnetic moments determine the magnetic properties of the materials.