Identify the effective length:
Velocity is normal to the shorter side (3 cm). This means the loop moves perpendicular to the 3 cm side.
The side that cuts through field lines as the loop exits = the side parallel to itself in the direction of motion = the shorter side = 3 cm = 0.03 m
Convert units:
l = 3 cm = 0.03 m (shorter side, effective cutting length)
v = 2 cm/s = 0.02 m/s
B = 0.3 T
Apply motional EMF formula:
\(\varepsilon = Bvl = 0.3 \times 0.02 \times 0.03\)
\(= 0.3 \times 6 \times 10^{-4} = 1.8 \times 10^{-4} \text{ V}\) ✓
First Law: Whenever magnetic flux through a circuit changes, an EMF is induced in it. Second Law: The magnitude of induced EMF = rate of change of magnetic flux: ε = −dΦ/dt. The negative sign indicates Lenz's law (induced current opposes the change causing it).
\(\varepsilon = -\dfrac{d\Phi}{dt}\)
Flux: Φ = BA cosθ
When a conductor of length l moves with velocity v perpendicular to a magnetic field B, the free electrons in the conductor experience a force F = qv × B, creating a potential difference (EMF) across the conductor:
\(\varepsilon = Bvl\) (when B, v, l all mutually perpendicular)
This is also consistent with Faraday's law: as the loop exits the field region, flux decreases at rate dΦ/dt = B × (rate of area leaving field) = B × v × l, giving ε = Bvl.
This is the key conceptual point in this problem. The loop has sides 8 cm (longer) and 3 cm (shorter). The velocity is normal to (perpendicular to) the shorter side — meaning the loop moves in the direction of the longer dimension. As the loop exits the field region:
📌 The side at the boundary between field and no-field region = the shorter side (3 cm)
📌 This 3 cm side sweeps out field lines as it moves with the loop
📌 The two longer sides (8 cm) move parallel to their own length — no cutting of field lines
📌 Effective length l = 3 cm = shorter side
The direction of induced current is such that it opposes the cause producing it. As the loop moves out of the magnetic field, flux decreases. By Lenz's law, the induced current must try to maintain the flux — so it flows in a direction to produce magnetic field in the same direction as the original field (using right-hand rule). This creates an opposing force on the loop, requiring external force to maintain constant velocity.
📌 ε = −dΦ/dt (Faraday's law)
📌 ε = Bvl (motional EMF, B⊥v⊥l)
📌 ε = NBAω sinωt (AC generator)
📌 ε_max = NBAω (peak EMF of AC generator)
📌 Lenz's law: induced current opposes flux change
Self-inductance L: EMF induced in a coil due to change in its own current: ε = −L(dI/dt). Unit: Henry (H). Mutual inductance M: EMF induced in secondary coil due to change in current in primary: ε₂ = −M(dI₁/dt). Transformer works on mutual inductance: V₁/V₂ = N₁/N₂ (ideal transformer).