The magnetic field is arise from the moving electric charges. A charge Q with velocity V giving rise to magnetic field B at point P in space, such that,
Where ' r' is the vector from Q to point P and where μ₀ it's called the permeability of vacuum or the magnetic constant, is define as 4π ×10⁻⁷ NC⁻² s² equation (1) it's valid especially only for a nonaccelerated charge moving with speed more less than that speed of light.
The vector B is called the magnetic flux density or magnetic induction. It was for more merly believed that the vector H was the most fundamental magnetic field vector, so H was called as a " magnetic field strength". It's now known that B is the fundamental magnetic vector. Equation (1) is in SI units with Q in coulomb's and B in teslas (T), where 1T = 1 NC⁻¹ m⁻¹ s.
The two electric charges +Q and - Q separated by a small distance b is constitute electric dipole. The electric dipole movement is defined as a vector from - Q to +Q with magnitudes Qb. For small planar loop of electric, current, it turns out that the magnetic field generated by the moving charges of the current is given by the same mathematical expression and that gives the electric field due to an electric dipole, except that the electric dipole moment is replaced by the magnetic dipole moment is m: m is a vector of magnitudes of IA, where l is the current flowing in a loop of area A. The direction m is perpendicular, plane of the current loop.
We can consider the magnetic dipole moments it's associated with a charge Q moving in a circle of speed ʋ with radius r. The current is flow per time, this circumstances of the circle is 2π r, and the times for one revolution is 2πr/ ʋ. Hence l= Qʋ/2πr. The magnitude of m is,
I m l = lA= (Qʋ/2πr) πr² = Qʋr/2 = Qrp/2m.................. (2)
Where m is the mass of the charged particles and p is it's linear momentum. But radius vector r is perpendicular to p, we have
.................... (3)
Where the definition of orbitals angular momentum L was used and the subscribt on "m" indicating that it arises from the orbital motions of the particles. Although we can derived equation (3) for the special cases of circular motion and it's validity is general. The electron, Q = - e, and the magnetic momentum due to it's orbital motion is,
The magnitude of L is gives and the magnitude of the orbital magnetic moment of electrons with orbitals - angular-momentum quantum number l is,
I mL l = eħ/2me [ l (l + 1)¹/₂ = m(mou)β [l (l + 1) ] ¹/₂.........(5)
The constant eħ/2mₑ is called the Bhor magneton m(mou)B :
m(mou)B ≡ eħ/ 2mₑ = 9.2740 × 10⁻²⁴ j/ T........................ (6)
Now, we can consider applying that external magnetic field to the hydrogen atom. The energy of interaction between a magnetic dipole "m" and an external magnetic field B can be shows to be,
EB = − m. B.............(7)
Using equation (4) we have
EB =e/2mₑ L. B...... (8)
We can take the ʐ direction, we have
EB = e/2mₑ B (Lₓᵢ + Lᵧⱼ + Lʐk ) . k = e/2mₑ BLʐ = m(mou) /ħ BLʐ
Where Lʐ is the ʐ component of orbitals angular momentum. Now we can replace Lʐ by the operator Lʐ to give the following additional terms in the Hamiltonian operator, and it's resulting from the external magnetic field,
Ĥʙ = μʙ ʙħ ⁻¹ L̂ʐ.......(9)
The Schrodinger equation for the hydrogen atom and it's magnetic field is,
( Ĥ + Hʙ ) ψ= Eψ.......... (10)
Where Ĥ is the hydrogen - atom Hamiltonian in the absence of external field. We readily verifying that the solutions of equation (10) are the complex hydrogenlike wave functions below,
( Ĥ + Ĥʙ ) R (r ) Yₗᵐ (θ, ø ) = Ĥ RYₗᵐ + μʙħ⁻¹ ʙL̂ ʐ RY ₗᵐ =
............ (11)
Thus, there's additional term μ ʙBm in the energy and the external magnetic field is removes the m is degeneracy. For obviously reasons, m is often called " magnetic quantum number" Actually, the observed energies shifting don't match the prediction of equation (11) because of the existence of electrons spin magnetic moment.
In quantum mechanics L lies on the surface of a cone model. A classical - mechanical treatment of the motion of L in applied magnetic fields showing that the field exerts a torque on mL, causing L to resolve about the direction of B at a constant frequency gives by l mL l B / 2π l L l, while the maintaining constant angle with B. This gyroscopic motion is called " precession. In quantum mechanics, a total complete specification of L is impossible. However, one finds that ⟨ L ⟩ processes about the field direction.
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