Problem 1: Motional EMF with Force

Given: Rod length l = 50 cm = 0.50 m, velocity v = 4 m/s, field B = 0.5 T, resistance R = 2 $\Omega$.

Step 1 — Motional EMF: $\varepsilon = Bvl = 0.5\,\text{T} \times 4\,\text{m/s} \times 0.50\,\text{m} = 1.0\,\text{V}$

Step 2 — Induced current: $I = \frac{\varepsilon}{R} = \frac{1.0\,\text{V}}{2\,\Omega} = 0.5\,\text{A}$

Step 3 — Force on rod (Lenz's law: opposing motion): $F = BIl = 0.5\,\text{T} \times 0.5\,\text{A} \times 0.50\,\text{m} = 0.125\,\text{N}$

Step 4 — Power check (energy conservation): $P_\text{external} = Fv = 0.125 \times 4 = 0.5\,\text{W}$ $P_\text{dissipated} = I^2 R = (0.5)^2 \times 2 = 0.5\,\text{W} \checkmark$