Electromagnetic Induction
Core concepts
Magnetic flux is (\Phi_B = \int \vec B\cdot d\vec A = BA\cos\theta). Faraday's law states that induced emf equals the negative rate of change of flux: (\varepsilon = -d\Phi_B/dt). The minus sign is Lenz's law: induced current opposes the change in flux that produced it. Flux changes come from changing (B), changing area, or changing orientation.
A motional emf arises when a conductor of length (L) moves at speed (v) through a field: (\varepsilon = BLv). Inductance (L) relates flux linkage to current, (\Phi = LI), and an inductor opposes changes in current with emf (\varepsilon = -L,dI/dt). RL circuits have time constant (\tau = L/R).
Traps
- Forgetting (\cos\theta): flux is maximum when (\vec B) is perpendicular to the loop's plane (parallel to the area vector), zero when parallel to the plane.
- Misapplying Lenz's law: the induced field opposes the change, not necessarily the original field. If flux is decreasing, the induced field points the same way.
- Confusing emf with current; emf exists even in an open loop, current requires a closed path.
- Ignoring that a stationary loop in a constant field has zero induced emf.
- Forgetting that an inductor opposes changes in current but acts like a wire in steady state.
Worked example
A square loop of side (0.2,\text{m}) sits in a field that increases uniformly from 0 to (0.5,\text{T}) in (2,\text{s}), field perpendicular to the loop. Area (= 0.04,\text{m}^2). (\Delta\Phi = 0.5 \times 0.04 = 0.02,\text{Wb}). Emf magnitude (= 0.02/2 = 0.01,\text{V}). Rule of thumb: compute flux at two instants, divide the difference by the time interval, then use Lenz's law to fix direction.