State Faraday's law of electromagnetic induction in integral form.

State Faraday's law of electromagnetic induction in integral form.

• A. ε = − dΦ_B/dt
• B. ε = dΦ_B/dt
• C. ε = Φ_B / dt
• D. ε = − ΔΦ_B

Answer: (A)

An induced electromotive force appears when the magnetic flux through a loop changes in time, and its direction follows Lenz’s law to oppose that change. The magnetic flux through the loop is Φ_B = ∫ B · dA, and the integral form of Faraday’s law is expressed as the line integral of the electric field around the loop: ∮ E · dl = - dΦ_B/dt. The EMF around the loop, often denoted ε, is this line integral, so ε = - dΦ_B/dt. This captures both that the emf is driven by how quickly the flux is changing and that the change is opposed by the induced current. The other expressions fail to match this relation because they miss the minus sign, representing the opposing direction required by Lenz’s law; or they use a simple division by dt or a finite change ΔΦ_B/dt instead of the instantaneous rate of change dΦ_B/dt. The correct form explicitly ties the emf to the instantaneous rate at which the magnetic flux through the loop is changing, with the sign enforcing opposition to that change.

An induced electromotive force appears when the magnetic flux through a loop changes in time, and its direction follows Lenz’s law to oppose that change. The magnetic flux through the loop is Φ_B = ∫ B · dA, and the integral form of Faraday’s law is expressed as the line integral of the electric field around the loop: ∮ E · dl = - dΦ_B/dt. The EMF around the loop, often denoted ε, is this line integral, so ε = - dΦ_B/dt. This captures both that the emf is driven by how quickly the flux is changing and that the change is opposed by the induced current.

The other expressions fail to match this relation because they miss the minus sign, representing the opposing direction required by Lenz’s law; or they use a simple division by dt or a finite change ΔΦ_B/dt instead of the instantaneous rate of change dΦ_B/dt. The correct form explicitly ties the emf to the instantaneous rate at which the magnetic flux through the loop is changing, with the sign enforcing opposition to that change.