Dimensional Formula of Magnetic Moment
Dimensional Formula: [M⁰ L² T⁰ I¹] or [L² I]
Derivation:
- Magnetic moment μ = Current × Area
- Dimension of current = [I]
- Dimension of area = [L²]
- Therefore, [μ] = [I] × [L²] = [L² I]
Alternative representation: [M⁰ L² T⁰ I¹ A⁰]
Detailed Explanations for Students
1. Basic Magnetic Moment (Current Loop)
The simplest definition relates to a current-carrying loop. When current flows through a loop, it creates a magnetic moment proportional to both the current and the loop’s area.
Example: A circular coil of radius 5 cm carrying 2 A current:
- A = πr² = π(0.05)² = 7.85×10⁻³ m²
- μ = 2 × 7.85×10⁻³ = 1.57×10⁻² A·m²
2. Spin-Only Magnetic Moment (Chemistry)
This is crucial for understanding transition metal complexes and their magnetic properties.
Formula: μ_s = √[n(n+2)] μ_B
Where n = number of unpaired electrons
Examples:
- Fe²⁺ (d⁶, 4 unpaired): μ = √[4(4+2)] = √24 = 4.90 μ_B
- Cu²⁺ (d⁹, 1 unpaired): μ = √[1(1+2)] = √3 = 1.73 μ_B
- Ti³⁺ (d¹, 1 unpaired): μ = √[1(1+2)] = √3 = 1.73 μ_B
3. Bohr Magneton
The Bohr magneton is the natural unit for expressing atomic magnetic moments.
Value: μ_B = 9.27×10⁻²⁴ J/T = 9.27×10⁻²⁴ A·m²
This represents the magnetic moment of an electron orbiting a nucleus.
4. Magnetic Dipole Moment
For bar magnets and magnetic dipoles:
- μ = pole strength × distance between poles
- Determines how strongly the dipole interacts with external fields
Frequently Asked Questions about Magnetic Moment Formulas
Q. What is the magnetic moment formula?
The magnetic moment (μ) is fundamentally defined as μ = I × A, where I is the current and A is the area enclosed by the current loop. For atomic systems, the spin-only formula μ = √[n(n+2)] μ_B is most commonly used, where n is the number of unpaired electrons.
Q. How do you calculate spin-only magnetic moment?
Use the formula μ_s = √[n(n+2)] μ_B, where n is the number of unpaired electrons in the atom or ion. First, determine the electronic configuration, count unpaired electrons, then substitute into the formula. The result is expressed in Bohr magnetons (μ_B).
Q. What is the dimensional formula of magnetic moment?
The dimensional formula is [M⁰ L² T⁰ I¹] or simply [L² I]. This is derived from magnetic moment = current × area, giving dimensions of length squared times current.
Q. What is the difference between spin and orbital magnetic moment?
Orbital magnetic moment arises from the electron’s motion around the nucleus (orbital angular momentum), while spin magnetic moment comes from the electron’s intrinsic spin. The spin-only formula is often used because orbital contributions are frequently “quenched” in transition metal complexes.
Q. Why is the Bohr magneton important?
The Bohr magneton (μ_B = 9.27×10⁻²⁴ J/T) is the natural unit of magnetic moment at the atomic scale. It represents the magnetic moment of one electron and provides a convenient scale for comparing atomic magnetic properties.
Q. How do you find the number of unpaired electrons from magnetic moment?
Rearrange the spin-only formula: n(n+2) = (μ/μ_B)². Solve this quadratic equation for n. For example, if μ = 3.87 μ_B, then n(n+2) = 15, giving n = 3 unpaired electrons.
Q. What is the magnetic moment of transition metals?
Transition metals have unpaired d-electrons that contribute to magnetic moment. Calculate using the spin-only formula based on d-electron configuration. For example, Fe³⁺ (d⁵) has 5 unpaired electrons: μ = √[5(7)] = 5.92 μ_B.
Q. What is the unit of magnetic moment?
The SI unit is ampere-meter squared (A·m²) or joule per tesla (J/T). In atomic physics, magnetic moment is expressed in Bohr magnetons (μ_B), where 1 μ_B = 9.27×10⁻²⁴ A·m².
Q. Why do we use the spin-only formula in chemistry?
In many transition metal complexes, orbital angular momentum is “quenched” due to the crystal field from ligands. This means orbital contributions to magnetic moment are suppressed, making the spin-only formula accurate for practical calculations.
Q. How does magnetic moment relate to paramagnetism?
Substances with unpaired electrons (and thus non-zero magnetic moment) are paramagnetic they are attracted to magnetic fields. The magnitude of the magnetic moment directly correlates with the strength of paramagnetic behavior. Diamagnetic substances have all electrons paired (μ = 0).
