07 / CHAPTER 11
Maxwell’s equations
See electricity and magnetism as one local conservation structure.
ANCHOR EQUATION
Maxwell synthesis
In symbols: ∇·E = ρ/ε₀
One of Maxwell’s local equations: charge is a source of electric field.
Induction is flux change, not field alone
A changing magnetic flux through a loop produces an electromotive force. The induced current opposes the change that created it, which is why generators resist their mechanical drive.
Differentiate magnetic flux
Magnetic flux is Φ_B = ∫B·dA. Faraday’s law gives ℰ = −dΦ_B/dt. For N turns, ℰ = −N dΦ_B/dt. The minus sign is Lenz’s law: reversing it would allow the induced current to reinforce the change and create energy from nothing.
A 0.020 m² loop in a perpendicular field increasing from 0 to 0.50 T in 0.10 s has ℰ = −AΔB/Δt = −0.020(0.50)/0.10 = −0.10 V.
No flux change means no induced emf
A large static field through a stationary loop produces no induction. A rotating loop, changing area, changing orientation, or changing field can each change flux and generate emf.
Work it: A 200-turn coil of area 4×10⁻⁴ m² sees B = 0.3 sin(100t) T perpendicular to it. Find the maximum induced emf.