What are Maxwell's equations in differential form? List the four equations.

What are Maxwell's equations in differential form? List the four equations.

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

Answer: (D)

These four equations encode how electric and magnetic fields respond to charges, currents, and changing fields. Gauss’s law in differential form tells you that the divergence of the electric field at a point equals the local charge density divided by ε0, so field lines originate from positive charges and terminate on negative charges. Faraday’s law says the curl of the electric field is minus the time rate of change of the magnetic field, capturing how a changing B field induces a circulating E field (the minus sign is tied to Lenz’s law). The condition that there are no magnetic monopoles is expressed by the fact that the divergence of B is zero, meaning magnetic field lines form closed loops rather than beginning or ending anywhere. Ampère–Maxwell’s law gives the curl of the magnetic field as produced both by electric currents and by changing electric fields, with the displacement current term μ0 ε0 ∂E/∂t ensuring consistency when charges move or fields vary in time. In vacuum, where there are no charges or currents, these equations still support electromagnetic wave propagation at speed c = 1/√(μ0 ε0). The form shown is the standard differential representation, correctly matching the signs and source terms.

These four equations encode how electric and magnetic fields respond to charges, currents, and changing fields. Gauss’s law in differential form tells you that the divergence of the electric field at a point equals the local charge density divided by ε0, so field lines originate from positive charges and terminate on negative charges. Faraday’s law says the curl of the electric field is minus the time rate of change of the magnetic field, capturing how a changing B field induces a circulating E field (the minus sign is tied to Lenz’s law). The condition that there are no magnetic monopoles is expressed by the fact that the divergence of B is zero, meaning magnetic field lines form closed loops rather than beginning or ending anywhere. Ampère–Maxwell’s law gives the curl of the magnetic field as produced both by electric currents and by changing electric fields, with the displacement current term μ0 ε0 ∂E/∂t ensuring consistency when charges move or fields vary in time. In vacuum, where there are no charges or currents, these equations still support electromagnetic wave propagation at speed c = 1/√(μ0 ε0). The form shown is the standard differential representation, correctly matching the signs and source terms.