Sources of Magnetic Fields Explained | Chapter 28 of University Physics

Chapter 28 explores how moving charges and currents create magnetic fields, and how materials respond to those fields. You’ll master the Biot–Savart Law for calculating field contributions, Ampère’s Law for symmetric configurations, and the role of magnetic susceptibility and permeability in different materials. Watch the full video summary for detailed derivations and visualizations.

Magnetic Field from a Moving Charge

A single charge moving with velocity v generates a magnetic field given by the equation:

B = (μ₀/4π) · [q v × r̂ / r²]

Here, μ₀ is the magnetic constant, q the charge, and r̂ the unit vector from the charge to the field point. The field lines form closed loops and obey the superposition principle.

Biot–Savart Law & Applications

The Biot–Savart Law extends this idea to current elements:

dB = (μ₀/4π) · [I dl × r̂ / r²]

By integrating over a wire, you find:

• Long straight wire: B = μ₀I/(2πr)
• Circular loop (on-axis): Bₓ = μ₀I a²/[2(a² + x²)^(3/2)]
• Multiple turns: multiply by the number of loops N.

Ampère’s Law & Symmetric Fields

Ampère’s Law relates the line integral of B around a closed path to the enclosed current:

∮B·dl = μ₀I_enc

Use it for highly symmetric cases:

• Straight conductor: recovers B = μ₀I/(2πr).
• Solenoid: B ≈ μ₀nI inside, nearly zero outside.
• Toroid: B = μ₀NI/(2πr) confined to the core.

Magnetic Materials & Permeability

Materials alter the applied field via magnetization M. The total field becomes:

B = μ₀(H + M) = μ₀μ_rH,

where μ_r is the relative permeability. Three classes exist:

• Paramagnetic (χ_m>0): weakly enhance B.
• Diamagnetic (χ_m<0): weakly oppose B.
• Ferromagnetic: strong, nonlinear response with hysteresis; used in permanent magnets.

Conclusion

By combining the Biot–Savart Law, Ampère’s Law, and an understanding of magnetic materials, you have the tools to calculate complex magnetic fields and predict material behavior under external fields. Practice applying these laws to solenoids, toroids, and circuits to deepen your mastery of magnetostatics.

For more chapter guides and practice problems, explore our full blog summaries.

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