Faraday's law states that the electromotive force around a closed loop equals what?

Faraday's law states that the electromotive force around a closed loop equals what?

• A. emf = dΦ_B/dt
• B. emf = - dΦ_B/dt
• C. emf = ∮ E · dl
• D. emf = Φ_B / dt

Answer: (B)

The essential idea is that a changing magnetic flux through a closed loop induces an electromotive force, and the direction is such that it opposes the change in flux (Lenz’s law). The magnetic flux through the surface bounded by the loop is Φ_B = ∫ B · dA. Faraday’s law then says that the induced emf around the loop is the negative rate of change of this flux: emf = - dΦ_B/dt. This negative sign is what enforces the opposition to the change in flux. Physically, this is tied to how changing magnetic fields create electric fields. Mathematically, curl E = - ∂B/∂t, and integrating around the loop with Stokes’ theorem gives emf = ∮ E · dl = - dΦ_B/dt. So the induced emf can be viewed either as the line integral of E around the loop or as the negative time derivative of the magnetic flux through the loop, with the minus sign encapsulating the opposing tendency described by Lenz’s law.

The essential idea is that a changing magnetic flux through a closed loop induces an electromotive force, and the direction is such that it opposes the change in flux (Lenz’s law). The magnetic flux through the surface bounded by the loop is Φ_B = ∫ B · dA. Faraday’s law then says that the induced emf around the loop is the negative rate of change of this flux: emf = - dΦ_B/dt. This negative sign is what enforces the opposition to the change in flux.

Physically, this is tied to how changing magnetic fields create electric fields. Mathematically, curl E = - ∂B/∂t, and integrating around the loop with Stokes’ theorem gives emf = ∮ E · dl = - dΦ_B/dt. So the induced emf can be viewed either as the line integral of E around the loop or as the negative time derivative of the magnetic flux through the loop, with the minus sign encapsulating the opposing tendency described by Lenz’s law.