Magnetism: Field Theory, Electromagnetic Forces, and Modern Applications

Fundamentals of Magnetism

Magnetism arises from moving electric charges and the intrinsic magnetic moments of elementary particles, creating dipole fields measured in teslas (T).

Key Equations

Lorentz Force Law

\[
\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})
\]
Where:

• \(\vec{F}\): Force vector (N)
• \(q\): Charge (C)
• \(\vec{v}\): Velocity vector (m/s)
• \(\vec{B}\): Magnetic field (T)

Magnetic Force on Current

\[
\vec{F} = I\vec{L} \times \vec{B}
\]
Special Case (θ=90°):
\[
F = BIL
\]

Advanced Concepts

Biot-Savart Law

\[
d\vec{B} = \frac{\mu_0}{4\pi}\frac{Id\vec{l} \times \hat{r}}{r^2}
\]
Where:

• \(\mu_0 = 4\pi \times 10^{-7}\) Tm/A
• \(d\vec{l}\): Current element

Ampere’s Law

\[
\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}
\]

Modern Applications

Electromechanical Systems

• Electric motors: 85-95% efficiency in modern designs
• Maglev trains: Field strengths up to 5T for levitation

Information Technology

• Hard disk drives: ~1T magnetic fields for data writing
• MRAM: Non-volatile memory using magnetic storage

Medical Technology

• MRI scanners: 1.5-7T superconducting magnets
• TMS: 1-2T pulsed fields for brain stimulation

Worked Example

Current-carrying wire:

• \(B = 0.3\) T
• \(I = 5\) A
• \(L = 1.5\) m
• θ = 90°

\[
F = (0.3)(5)(1.5)\sin90° = 2.25 \, \text{N}
\]

Common Errors

1. Misapplying right-hand rule for force direction
2. Confusing magnetic field units (1T = 10⁴ Gauss)
3. Neglecting relativistic effects in high-velocity cases

Practice Problems

1. A proton (\(q = 1.6 \times 10^{-19}\) C) moves at \(2 \times 10^6\) m/s perpendicular to a 0.5T field:
   • Calculate the magnetic force
   • Determine the radius of its circular path
2. Derive the relationship \(B = \mu_0 nI\) for solenoid fields
3. Compare permanent magnets vs electromagnets for MRI applications.