Which Maxwell equation in differential form relates the curl of B to current density and the time rate of change of E (the Ampere-Maxwell law)?

Which Maxwell equation in differential form relates the curl of B to current density and the time rate of change of E (the Ampere-Maxwell law)?

• A. ∇×B = μ0 J + μ0 ε0 ∂E/∂t
• B. ∇×E = −∂B/∂t
• C. ∇·E = ρ/ε0
• D. ∇·B = 0

Answer: (A)

The curl of the magnetic field is driven by circulating electric effects, so the differential form that ties ∇×B to currents and changing electric fields is Ampere-Maxwell law. It states that ∇×B equals μ0 times the current density J plus μ0 ε0 times the time rate of change of the electric field ∂E/∂t. The first term captures the magnetic field produced by moving charges, while the second term—the displacement current—accounts for changing electric fields even when no physical current flows (such as during capacitor charging). This combination explains how magnetic fields can form loops around both actual currents and changing electric fields. The other equations describe different phenomena: curl E equals −∂B/∂t is Faraday’s law of induction, linking changing magnetic fields to induced electric fields; divergence E equals ρ/ε0 is Gauss’s law for electricity, relating the electric field to charge density; and divergence B equals zero is Gauss’s law for magnetism, reflecting that there are no magnetic monopoles.

The curl of the magnetic field is driven by circulating electric effects, so the differential form that ties ∇×B to currents and changing electric fields is Ampere-Maxwell law. It states that ∇×B equals μ0 times the current density J plus μ0 ε0 times the time rate of change of the electric field ∂E/∂t. The first term captures the magnetic field produced by moving charges, while the second term—the displacement current—accounts for changing electric fields even when no physical current flows (such as during capacitor charging). This combination explains how magnetic fields can form loops around both actual currents and changing electric fields.

The other equations describe different phenomena: curl E equals −∂B/∂t is Faraday’s law of induction, linking changing magnetic fields to induced electric fields; divergence E equals ρ/ε0 is Gauss’s law for electricity, relating the electric field to charge density; and divergence B equals zero is Gauss’s law for magnetism, reflecting that there are no magnetic monopoles.