A conducting rod of length L moves with velocity v parallel to a uniform magnetic field B. The induced emf is zero because the motion is not perpendicular to B. True or False?

• A. True
• B. False
• C. ε = B L v
• D. ε = B L^2 / v

Answer: (A)

Motional emf comes from the magnetic force on moving charges, which is q(v × B). Its magnitude scales with how much of the velocity is perpendicular to the field. If the rod is moving with velocity parallel to the magnetic field, v × B is zero everywhere along the rod, so there is no magnetic force to separate charges and no potential difference develops. Hence the induced emf is zero. The reason given in the statement—that the emf is zero because the motion is not perpendicular to B—captures the right result, but the precise reason is that the velocity is parallel to B, making the cross product vanish. In general, the emf would be ε = ∫(v × B) · dl = B L v sinθ, where θ is the angle between v and B; this gives ε = 0 when θ = 0 (parallel) and ε = B L v when θ = 90° (perpendicular). The other formulas do not apply here.