How is an emf induced in a conducting rod moving with velocity v perpendicular to a uniform magnetic field B (motional emf)?

How is an emf induced in a conducting rod moving with velocity v perpendicular to a uniform magnetic field B (motional emf)?

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

Answer: (A)

When a conducting rod moves through a magnetic field, charges inside feel the magnetic force F = q v × B. If the velocity is perpendicular to B, this magnetic force has a constant magnitude q v B and acts along the length of the rod, pushing positive and negative charges toward opposite ends. This separation creates an electric potential difference across the ends—the motional emf. To get its magnitude, consider the force per unit charge along the rod: (v × B) has magnitude vB, acting along the rod’s length L. The potential difference accumulates over the length, so the emf is ε = ∫ (v × B) · dl = (vB) ∫ dl = B L v. Thus, the induced emf scales with the magnetic field strength, the rod’s length, and the velocity: ε = B L v when v is perpendicular to B. If the circuit is closed, this emf drives a current; the direction is given by v × B for positive charges.

When a conducting rod moves through a magnetic field, charges inside feel the magnetic force F = q v × B. If the velocity is perpendicular to B, this magnetic force has a constant magnitude q v B and acts along the length of the rod, pushing positive and negative charges toward opposite ends. This separation creates an electric potential difference across the ends—the motional emf.

To get its magnitude, consider the force per unit charge along the rod: (v × B) has magnitude vB, acting along the rod’s length L. The potential difference accumulates over the length, so the emf is ε = ∫ (v × B) · dl = (vB) ∫ dl = B L v.

Thus, the induced emf scales with the magnetic field strength, the rod’s length, and the velocity: ε = B L v when v is perpendicular to B. If the circuit is closed, this emf drives a current; the direction is given by v × B for positive charges.