Which expression represents Ampere's law with Maxwell's correction in integral form?

Which expression represents Ampere's law with Maxwell's correction in integral form?

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

Answer: (B)

Ampere’s law with Maxwell’s correction says the line integral of the magnetic field around a closed loop depends not only on the current that passes through the loop but also on the changing electric flux through the loop’s surface. The correct integral form is ∮ B · dl = μ0 I_enclosed + μ0 ε0 dΦ_E/dt. This shows both contributions: the actual current and the displacement current term from the changing electric flux. The μ0 factor multiplies the entire sum, so the displacement term is μ0 ε0 dΦ_E/dt. This correction is essential for situations like charging a capacitor, where there is a changing electric field between plates but no physical current through the dielectric. The other forms miss or misplace parts of this expression. Without the displacement term, the law can fail in time-varying situations; with the incorrect factor, the units don’t match; and an integral like ∮ E × dl isn’t the correct magnetic-field circulation form.

Ampere’s law with Maxwell’s correction says the line integral of the magnetic field around a closed loop depends not only on the current that passes through the loop but also on the changing electric flux through the loop’s surface. The correct integral form is

∮ B · dl = μ0 I_enclosed + μ0 ε0 dΦ_E/dt.

This shows both contributions: the actual current and the displacement current term from the changing electric flux. The μ0 factor multiplies the entire sum, so the displacement term is μ0 ε0 dΦ_E/dt. This correction is essential for situations like charging a capacitor, where there is a changing electric field between plates but no physical current through the dielectric.

The other forms miss or misplace parts of this expression. Without the displacement term, the law can fail in time-varying situations; with the incorrect factor, the units don’t match; and an integral like ∮ E × dl isn’t the correct magnetic-field circulation form.