Ampere's Law is one of the fundamental laws in electromagnetism that relates the magnetic field to the electric current. It states that the circulation of the magnetic field, commonly referred to as the curl of the magnetic field (H), around a closed loop is equal to the current passing through the loop multiplied by a constant.

Curl of H is same as the current density (Option D):

- Ampere's Law mathematically states that the circulation of the magnetic field, represented by the curl of H (∇ x H), around a closed loop is equal to the total current passing through the loop. This is given by the equation:
∮(H · dl) = ∫(J · dA)

- Here, ∮(H · dl) represents the circulation of the magnetic field around a closed loop, which is the integral of the dot product of the magnetic field (H) and an infinitesimal element of the loop (dl). ∫(J · dA) represents the total current passing through the loop, which is the integral of the dot product of the current density (J) and an infinitesimal element of area (dA) perpendicular to the current.

- By using the divergence theorem, which relates a surface integral to a volume integral, we can rewrite the equation as:
∇ x H = J

- The left-hand side of the equation represents the curl of the magnetic field (H), while the right-hand side represents the current density (J). Therefore, the curl of H is indeed equal to the current density, confirming Option D as the correct answer.

- This relationship is crucial in analyzing and predicting the behavior of magnetic fields generated by electric currents. It allows us to calculate magnetic fields around conductors, solenoids, and other current-carrying structures.

In summary, Ampere's Law states that the curl of the magnetic field (H) around a closed loop is equal to the current passing through the loop. This relationship, represented by ∇ x H = J, confirms that the curl of H is indeed equal to the current density.