For a rod of length L moving with speed v perpendicular to a uniform magnetic field B, the induced emf is?

For a rod of length L moving with speed v perpendicular to a uniform magnetic field B, the induced emf is?

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

Answer: (C)

The key idea here is motional emf: charges in a conductor moving through a magnetic field experience a magnetic force F = q(v × B). This force pushes charges along the length of the rod, creating a potential difference between its ends. The emf is the line integral of this force per unit charge along the rod: ε = ∫ (v × B) · dl. If the velocity is perpendicular to B, the magnitude of v × B is vB. If the rod is oriented along the direction of this force, dl points in the same direction as v × B, so (v × B) · dl = vB dl. Integrating over the rod’s length L gives ε = vB L. Thus the induced emf is B L v.

The key idea here is motional emf: charges in a conductor moving through a magnetic field experience a magnetic force F = q(v × B). This force pushes charges along the length of the rod, creating a potential difference between its ends. The emf is the line integral of this force per unit charge along the rod: ε = ∫ (v × B) · dl.

If the velocity is perpendicular to B, the magnitude of v × B is vB. If the rod is oriented along the direction of this force, dl points in the same direction as v × B, so (v × B) · dl = vB dl. Integrating over the rod’s length L gives ε = vB L.

Thus the induced emf is B L v.