Maxwell's First Equation Point Form Explained

Maxwell's equations are fundamental laws that describe the behavior of electric and magnetic fields. The question asks for the point form of Maxwell's first equation, specifically for time-varying fields.

Maxwell's first equation is essentially Gauss's Law for Electricity. It relates the divergence of the electric displacement field ($\overrightarrow D$) to the volume charge density ($\rho$). The point form of this law is expressed as:

\(\overrightarrow \nabla \cdot \overrightarrow D = \rho \)

This equation states that the net outward flux of the electric displacement field through any closed surface is equal to the total electric charge enclosed within that surface. Importantly, this relationship holds true irrespective of whether the fields are static or time-varying.

Analyzing the Options

Let's examine why the other options are not the correct point form of Maxwell's first equation:

• Option 2: \(\overrightarrow \nabla \times \overrightarrow D = \rho \) - This is incorrect. The curl ($\overrightarrow \nabla \times$) of the electric displacement field ($\overrightarrow D$) is related to current density and the rate of change of the electric field, not directly to charge density ($\rho$).
• Option 3: \(\overrightarrow \nabla \cdot \overrightarrow D = \overrightarrow J + \frac{{\partial \overrightarrow D}}{{\partial t}}\) - This equation resembles Ampère's Law with Maxwell's addition, but it uses divergence instead of curl and applies to the magnetic field ($\overrightarrow B$ or $\overrightarrow H$), not the electric displacement field ($\overrightarrow D$) in this manner for Gauss's Law. The correct form relating divergence of D to charge density does not include current density ($\overrightarrow J$) or the time derivative of D.
• Option 4: \(\overrightarrow \nabla \cdot \overrightarrow D = 0\) - This equation represents Gauss's Law for Electricity only in regions where the volume charge density ($\rho$) is zero. It is a special case, not the general form of the equation.

Therefore, the correct point form of Maxwell's first equation, applicable to both static and time-varying fields, is \(\overrightarrow \nabla \cdot \overrightarrow D = \rho \).