Ampere's circuit law for time varying fields is given by:

Explanation:

Ampere's Circuital Law for Time-Varying Fields

Definition: Ampere's Circuital Law is a fundamental relationship in electromagnetics that relates the magnetic field circulating around a closed path to the electric current passing through the surface enclosed by that path. For time-varying fields, the law is modified to include the contribution of displacement current, which accounts for the changing electric field.

Mathematical Expression:

For time-varying fields, Ampere's Circuital Law is expressed as:

\[\nabla \times \overline{\mathrm{H}} = \overline{\mathrm{J}_{\mathrm{C}}} + \overline{\mathrm{J}_{\mathrm{D}}}\]

where:

• \(\nabla \times \overline{\mathrm{H}}\): The curl of the magnetic field intensity.
• \(\overline{\mathrm{J}_{\mathrm{C}}}\): The conduction current density.
• \(\overline{\mathrm{J}_{\mathrm{D}}}\): The displacement current density.

Displacement Current: The concept of displacement current was introduced by James Clerk Maxwell to address the inconsistency in Ampere's Law for time-varying fields. It is given by:

\[\overline{\mathrm{J}_{\mathrm{D}}} = \epsilon_0 \frac{\partial \overline{\mathrm{E}}}{\partial t}\]

where:

• \(\epsilon_0\): The permittivity of free space.
• \(\frac{\partial \overline{\mathrm{E}}}{\partial t}\): The rate of change of the electric field.

The term displacement current is not a physical current but an apparent current that arises from the time-varying electric field.

Correct Option Analysis:

The correct expression for Ampere's Circuital Law in the presence of time-varying fields is:

\[\nabla \times \overline{\mathrm{H}} = \overline{\mathrm{J}_{\mathrm{C}}} + \overline{\mathrm{J}_{\mathrm{D}}}\]

Thus, the correct answer is Option 2.

Additional Information

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

Option 1: \(\nabla \times \overline{\mathrm{H}} = \overline{\mathrm{J}_{\mathrm{C}}} - \overline{\mathrm{J}_{\mathrm{D}}}\)

This option incorrectly subtracts the displacement current density \(\overline{\mathrm{J}_{\mathrm{D}}}\) from the conduction current density \(\overline{\mathrm{J}_{\mathrm{C}}}\). In Ampere's Circuital Law for time-varying fields, these two terms are additive because both contribute to the curl of the magnetic field intensity.

Option 3: \(\nabla \times \overline{\mathrm{H}} = \mathrm{J}_{\mathrm{C}} + \mathrm{J}_{\mathrm{D}}\)

This option has the correct additive relationship but uses scalar current densities (\(\mathrm{J}_{\mathrm{C}}\) and \(\mathrm{J}_{\mathrm{D}}\)) instead of vector current densities (\(\overline{\mathrm{J}_{\mathrm{C}}}\) and \(\overline{\mathrm{J}_{\mathrm{D}}}\)). Since current density is a vector quantity, this representation is incomplete and, therefore, incorrect.

Option 4: \(\nabla \times \overline{\mathrm{H}} = \mathrm{J}_{\mathrm{C}} - \mathrm{J}_{\mathrm{D}}\)

This option is incorrect for two reasons: it subtracts the displacement current density instead of adding it, and it uses scalar current densities instead of vector current densities. Both aspects deviate from the proper formulation of Ampere's Circuital Law for time-varying fields.

Conclusion:

Ampere's Circuital Law is a cornerstone of electromagnetics, especially in the context of Maxwell's equations. For time-varying fields, the inclusion of displacement current ensures the consistency of the law and its applicability to scenarios involving changing electric and magnetic fields. The correct mathematical expression:

\[\nabla \times \overline{\mathrm{H}} = \overline{\mathrm{J}_{\mathrm{C}}} + \overline{\mathrm{J}_{\mathrm{D}}}\]

highlights the contributions of both conduction current and displacement current to the magnetic field. This understanding is critical for analyzing electromagnetic wave propagation, antenna design, and other advanced topics in electromagnetics.