Understanding Maxwell's Equations for Static Fields

Maxwell's equations are a set of fundamental equations that describe the behavior of electric and magnetic fields. For a static electromagnetic field in a linear homogeneous medium, these equations simplify. The question asks us to identify which of the given options is *not* one of these static Maxwell equations or a direct consequence thereof.

Static Maxwell's Equations (Differential Form)

The four fundamental Maxwell's equations in differential form are:

• Gauss's Law for Electricity: \(\nabla \cdot \overrightarrow D = \rho_f\)
• Gauss's Law for Magnetism: \(\nabla \cdot \overrightarrow B = 0\)
• Faraday's Law of Induction: \(\nabla \times \overrightarrow E = - \frac{\partial \overrightarrow B}{\partial t}\)
• Ampère-Maxwell Law: \(\nabla \times \overrightarrow H = \overrightarrow J_f + \frac{\partial \overrightarrow D}{\partial t}\)

For static electromagnetic fields, the time-dependent terms are zero (\(\frac{\partial \overrightarrow B}{\partial t} = 0\) and \(\frac{\partial \overrightarrow D}{\partial t} = 0\)). The equations become:

• \(\nabla \cdot \overrightarrow D = \rho_f\)
• \(\nabla \cdot \overrightarrow B = 0\)
• \(\nabla \times \overrightarrow E = 0\)
• \(\nabla \times \overrightarrow H = \overrightarrow J_f\)

In a linear homogeneous medium, we also use the constitutive relations: \(\overrightarrow D = \epsilon \overrightarrow E\) and \(\overrightarrow B = \mu \overrightarrow H\), where \(\epsilon\) and \(\mu\) are constants.

Analyzing the Options

Option 1 Analysis: \(\nabla \cdot \overrightarrow B = 0\)

This equation represents Gauss's Law for Magnetism. It signifies the absence of magnetic monopoles, meaning magnetic field lines always form closed loops. This is one of the four fundamental Maxwell's equations and is valid for static fields. Therefore, it is a Maxwell equation for a static field.

Option 2 Analysis: \( \nabla \times \overrightarrow D = 0\)

Let's analyze this equation. For static fields, Faraday's Law simplifies to \(\nabla \times \overrightarrow E = 0\). Using the constitutive relation \(\overrightarrow D = \epsilon \overrightarrow E\) for a linear homogeneous medium, we can substitute:

\(\nabla \times \overrightarrow D = \nabla \times (\epsilon \overrightarrow E)\)

Since \(\epsilon\) is constant in a homogeneous medium, we can pull it out of the curl:

\(\nabla \times \overrightarrow D = \epsilon (\nabla \times \overrightarrow E)\)

Substituting \(\nabla \times \overrightarrow E = 0\):

\(\nabla \times \overrightarrow D = \epsilon \cdot 0 = 0\)

This equation, \(\nabla \times \overrightarrow D = 0\), is a correct consequence derived from Maxwell's equations and constitutive relations for static fields. However, it is not one of the four *fundamental* Maxwell's equations themselves; it is a derived relationship.

Option 3 Analysis: \(\oint {\overrightarrow B \cdot dl = ({\mu _0}/4\pi )I} \)

This equation is given in integral form. The standard integral form of Ampère's Law relates the line integral of the magnetic field \(\overrightarrow B\) around a closed loop to the total current \((I_{enc})\) enclosed by the loop. In a medium with permeability \(\mu\), it is:

\(\oint {\overrightarrow B \cdot d\overrightarrow l} = \mu I_{enc}\)

The equation provided uses \(\mu_0/4\pi\), which is a constant associated with the Biot-Savart law for a current element, not typically found in the integral form of Ampère's Law. Furthermore, for a general medium, \(\mu\) should be used instead of \(\mu_0\). Therefore, this equation is not a correct general representation of Ampère's Law for a static field in a medium.

Option 4 Analysis: \(\nabla(\nabla \cdot \overrightarrow A)- (\nabla^2 \overrightarrow A)= {\mu _0}J\)

This equation can be simplified using the vector identity \(\nabla \times (\nabla \times \overrightarrow A) = \nabla(\nabla \cdot \overrightarrow A) - \nabla^2 \overrightarrow A\). Thus, the equation becomes:

\(\nabla \times (\nabla \times \overrightarrow A) = \mu_0 J\)

Since the magnetic field is defined as \(\overrightarrow B = \nabla \times \overrightarrow A\), this equation implies:

\(\nabla \times \overrightarrow B = \mu_0 J\)

This is the differential form of Ampère's Law for static fields. However, it is strictly valid for vacuum, where \(\mu = \mu_0\). For a general linear homogeneous medium, the correct form is \(\nabla \times \overrightarrow B = \mu J\). Therefore, this equation is not universally correct for any static field in a linear homogeneous medium.

Conclusion

Based on the analysis, Option 1 is a fundamental Maxwell equation. Options 3 and 4 are either incorrectly formulated or specific to vacuum, not a general medium. However, the question asks which equation is *not* Maxwell's equation. Option 2, \(\nabla \times \overrightarrow D = 0\), while a valid consequence, is derived using constitutive relations and is not considered one of the four primary Maxwell's equations. Therefore, it is the equation that does not fit the definition of being one of Maxwell's fundamental equations.