Faraday's Law of Induction:
\[ \mathcal{E} = -N \frac{d\Phi_B}{dt} \]
Where:
- \(N\) = Number of coil turns
- \(\Phi_B\) = Magnetic flux \(= BA\cos\theta\)
- \(B\) = Magnetic field strength
- \(A\) = Coil cross-sectional area
- \(\theta\) = Angle between B and A
The negative sign indicates Lenz's Law: the induced current creates a magnetic field that opposes the change in flux.
Advanced Concepts:
- Magnetic Field Strength: Decreases with square of distance (\(B \propto \frac{1}{r^2}\))
- Flux Change: Maximum when magnet moves perpendicular to coil axis
- Induced Current: Proportional to rate of flux change
- Lenz's Law: Induced current opposes flux change
\[ B(r) = \frac{B_0}{(r^2 + a^2)} \]
Where \(a\) is the characteristic length of the magnet.