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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
![\[ \vec{F} = q(\vec{E} + \vec{v} \times \vec{B}) \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-228ff7ab56ccc5009777b080403eb366_l3.png "Rendered by QuickLaTeX.com")
Where:
: Force vector (N)
: Charge (C)
: Velocity vector (m/s)- : Magnetic field (T)
: Force vector (N)
: Charge (C)
: Velocity vector (m/s)
: Magnetic field (T)Magnetic Force on Current
![\[ \vec{F} = I\vec{L} \times \vec{B} \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-1bcba799f2b32cc8427933edec3e5989_l3.png "Rendered by QuickLaTeX.com")
Special Case (θ=90°):
![\[ F = BIL \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-97e9052e55ea91c4e37477bb0624408d_l3.png "Rendered by QuickLaTeX.com")
Advanced Concepts
Biot-Savart Law
![\[ d\vec{B} = \frac{\mu_0}{4\pi}\frac{Id\vec{l} \times \hat{r}}{r^2} \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-e24cdb5943fcbfef78aceb430c91c007_l3.png "Rendered by QuickLaTeX.com")
Where:
Tm/A
: Current element
Tm/A
: Current elementAmpere’s Law
![\[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-86d6d495841eb2de1510852d96b49407_l3.png "Rendered by QuickLaTeX.com")
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:
T
A
m- θ = 90°
T
A
m ![\[ F = (0.3)(5)(1.5)\sin90° = 2.25 \, \text{N} \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-f5fa4c6f8f6b4149eda95ff735d79c88_l3.png "Rendered by QuickLaTeX.com")
Common Errors
- Misapplying right-hand rule for force direction
- Confusing magnetic field units (1T = 10⁴ Gauss)
- Neglecting relativistic effects in high-velocity cases
Practice Problems
- A proton (
C) moves at 
m/s perpendicular to a 0.5T field:- Calculate the magnetic force
- Determine the radius of its circular path
- Derive the relationship
for solenoid fields - Compare permanent magnets vs electromagnets for MRI applications
C) moves at 
m/s perpendicular to a 0.5T field:
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