Which of the following formulae gives Maxwell's first equation?

Explanation:

Maxwell's First Equation

Definition: Maxwell's first equation, also known as Gauss's law for electricity, describes the relationship between electric charge and electric flux density. It states that the divergence of the electric flux density (\( \overline{D} \)) is equal to the volume charge density (\( \rho_{v} \)). This fundamental equation is derived from the principles of electrostatics and is expressed mathematically as:

\(\rm \operatorname{div} \overline{D} =\rho_{v}\)

This equation is essential in understanding how electric charges create electric fields and how they interact with their surroundings.

Understanding the Terms:

• \(\rm \overline{D}\) (Electric Flux Density): Represents the amount of electric flux passing through a unit area. It is related to the electric field (\(\rm \overline{E}\)) and the permittivity of the medium (\(\varepsilon\)) as \(\rm \overline{D} = \varepsilon \overline{E}\).
• \(\rho_{v}\) (Volume Charge Density): Represents the amount of electric charge per unit volume.
• \(\rm \operatorname{div}\): The divergence operator calculates the net flux leaving a point in space.

Explanation of Maxwell's First Equation:

Maxwell's first equation states that the divergence of the electric flux density (\(\rm \operatorname{div} \overline{D}\)) at a point is equal to the volume charge density (\(\rho_{v}\)) at that point. Physically, it implies that electric charges act as sources or sinks of electric flux. Positive charges emit flux, while negative charges absorb flux.

In integral form, Gauss's law can be expressed as:

\(\int_{S} \overline{D} \cdot \overline{n} \, dS = Q_{enc}\)

Where:

• \(S\): A closed surface.
• \(\overline{n}\): Unit normal vector to the surface.
• \(Q_{enc}\): The total charge enclosed within the surface \(S\).

This equation highlights that the total electric flux through a closed surface is proportional to the charge enclosed within that surface.

Correct Option Analysis:

The correct option is:

Option 4: \(\rm \operatorname{div} \overline{D} =\rho_{v}\)

This option correctly represents Maxwell's first equation in its differential form. It accurately describes the relationship between the divergence of the electric flux density (\(\rm \operatorname{div} \overline{D}\)) and the volume charge density (\(\rho_{v}\)). This is a foundational expression in electromagnetism and forms the basis for understanding electric fields and their interaction with charges.

Important Information

To further understand the analysis, let’s evaluate the other options:

Option 1: \(\rm \operatorname{div} D =\overline{\rho_{v}}\)

This option uses the term \(D\) without the vector notation (\(\overline{D}\)). In electromagnetism, the electric flux density is a vector quantity, and its divergence is calculated with respect to its components. Therefore, the omission of the vector notation makes this representation incomplete and incorrect.

Option 2: \(\rm \operatorname{div} \overline{D} =\overline{\rho_{v}}\)

This option incorrectly represents the volume charge density (\(\rho_{v}\)) as a vector quantity (\(\overline{\rho_{v}}\)). However, \(\rho_{v}\) is a scalar quantity representing the amount of charge per unit volume. Since divergence results in a scalar value, this representation is incorrect.

Option 3: \(\rm div D = \rho_{v}\)

This option omits the vector notation for the electric flux density (\(D\)). While the equation may appear correct in form, the lack of vector notation for \(D\) makes it incomplete, as \(D\) is a vector field. Proper representation requires the use of \(\rm \overline{D}\) to signify the vector nature of the electric flux density.

Conclusion:

Maxwell's first equation is a cornerstone of electromagnetism, describing the interaction between electric charges and the electric fields they produce. The correct representation of this equation in differential form is \(\rm \operatorname{div} \overline{D} =\rho_{v}\), as stated in option 4. This equation highlights the fundamental relationship between the divergence of the electric flux density and the volume charge density, providing essential insights into electric field behavior. Understanding the correct formulation of Maxwell's equations is critical for studying and applying electromagnetic principles in various engineering and scientific applications.