In the absence of changing electric flux, Ampere's law reduces to which relation?

• A. ∮ B · dl = μ0 I_enclosed
• B. ∮ B · dl = μ0 I_enclosed + μ0 ε0 dΦ_E/dt
• C. ∮ B · dl = μ0 ε0 dΦ_E/dt
• D. ∮ B · dl = 0

Answer: (A)

When you take the line integral of the magnetic field around a closed loop, Ampere’s law relates it to the current that passes through the loop plus the effect of any changing electric field. The full relation is ∮ B · dl = μ0 I_enclosed + μ0 ε0 dΦ_E/dt. If the electric flux through the surface bounded by the loop does not change with time, the displacement current term μ0 ε0 dΦ_E/dt drops to zero. What remains is ∮ B · dl = μ0 I_enclosed, meaning the magnetic field looping around the path is produced solely by the actual current threading the loop. This is why, in the absence of a changing electric flux, the line integral reduces to μ0 times the enclosed current. If the electric flux were changing, the displacement term would contribute and the full expression would be needed.