Consider the following statements regarding Maxwell’s equation in differential form:

a) For free space: ∇ × H = (σ + jωε) E

b) For free space: ∇ ⋅ D ≃ ρ

c) For steady current: ∇ × H = J

d) For static electric field: ∇ ⋅ D = ρ

What of the above statements are correct?

Maxwell’s equation for time-varying electromagnetic fields are:

1) In Differential or point form:

\(\vec \nabla \cdot \vec D = {ρ _v}\)   ---(1)

This is valid for both static and time-varying fields.

(Statement (d) is correct)

In Integral form:

\(\mathop \oint \nolimits_s \vec D \cdot d\vec s = \mathop \smallint \nolimits_v {ρ _v}dV\)

2) \(\vec \nabla \times \vec H = \vec J + \frac{{\partial \vec D}}{{\partial t}}\)   ---(2)

For steady current, the change of field with time is zero, i.e.

\( \frac{{\partial \vec D}}{{\partial t}}=0\)

∴ \(\vec \nabla \times \vec H = \vec J \)

(Statement (c) is correct)

With J = σ E and D = ϵ E, Equation (2) becomes:

\(\vec \nabla \times \vec H = σ \vec E + \frac{{ \epsilon\partial \vec E}}{{\partial t}}\)

In time-harmonic form, this can be written as:

∇ × H = (σ + jωε) E

For free space, σ = 0, and ρ = 0:

∇ × H = jωε E (Statement (a) is incorrect)

Similarly, for free space, Equation (1) becomes:

∇ ⋅ D ≃ 0 (Statement (b) is incorrect)