Understanding the Differential Form of Electromagnetism Laws

The question asks to identify the fundamental law of electromagnetism represented by the differential equation $\nabla \times H = J$. This equation involves the curl operator ($\nabla \times$), the magnetic field intensity ($H$), and the current density ($J$). Understanding what each term means and how it relates to the behavior of electromagnetic fields is key to solving this.

Decoding the Equation: $\nabla \times H = J$

• $\nabla \times$: This is the curl operator. In electromagnetism, the curl of a vector field tells us about the rotation or circulation of that field at a point. A non-zero curl means the field "swirls" around that point.
• $H$: This represents the magnetic field intensity. It is a vector field that describes how a magnetic field affects a region.
• $J$: This is the current density vector. It represents the amount of electric current flowing per unit cross-sectional area and its direction.

So, the equation $\nabla \times H = J$ mathematically states that the curl of the magnetic field intensity at any point is equal to the current density at that point. In simpler terms, electric currents are the source of magnetic fields that swirl around them.

Relating to Fundamental Laws

Let's look at the given options and their differential forms:

1. Gauss’s law: Gauss’s law for electricity relates electric fields to electric charges. Its differential form is $\nabla \cdot D = \rho_v$, where $D$ is the electric displacement field and $\rho_v$ is the volume charge density. Gauss’s law for magnetism states there are no magnetic monopoles; its differential form is $\nabla \cdot B = 0$, where $B$ is the magnetic flux density. Neither matches $\nabla \times H = J$.
2. Ampere’s circuital law: Ampere’s circuital law, in its original form for static magnetic fields, relates the circulation of the magnetic field intensity around a closed loop to the total current passing through the loop. The integral form is $\oint_L H \cdot dl = I_{enclosed}$. Using Stokes' theorem, this integral form can be converted to a differential form: $\nabla \times H = J$. Maxwell later added the displacement current term ($\partial D / \partial t$) to make it applicable to time-varying fields, resulting in the Ampere-Maxwell equation: $\nabla \times H = J + \frac{\partial D}{\partial t}$. However, the equation $\nabla \times H = J$ itself is the differential form of Ampere's circuital law, especially for static conditions or when displacement current is negligible.
3. Poisson’s equation: Poisson’s equation is a second-order partial differential equation that arises from Gauss's law and the definition of electric potential ($\nabla V = -E$). For instance, in electrostatics, it is $\nabla^2 V = -\rho_v / \epsilon_0$, where $V$ is the electric potential and $\epsilon_0$ is the vacuum permittivity. It does not match the given equation.
4. Laplace’s equation: Laplace’s equation is a special case of Poisson’s equation where the charge density is zero ($\rho_v = 0$). Its form is $\nabla^2 V = 0$. It does not match the given equation.

Comparing the given equation $\nabla \times H = J$ with the differential forms of the options, it directly matches the differential form of Ampere's circuital law (static case).

Conclusion

Based on the differential forms of the fundamental laws, the equation $\nabla \times H = J$ is the differential form of Ampere’s circuital law, specifically in the static case or when the displacement current term is ignored.

Revision Table: Key Electromagnetic Equations

Note that the equation given in the question, $\nabla \times H = J$, corresponds to the Ampere-Maxwell law when the time-varying electric displacement term ($\frac{\partial D}{\partial t}$) is zero, which is the case for static or steady currents.

Additional Information on Vector Operators in Electromagnetism

Vector calculus is essential for understanding the differential forms of electromagnetic laws. The operators used are gradient, divergence, and curl.

• Gradient ($\nabla$): Applies to a scalar field and produces a vector field pointing in the direction of the steepest increase of the scalar field. Example: $\nabla V$ gives the electric field $E$ from the electric potential $V$ ($E = -\nabla V$).
• Divergence ($\nabla \cdot$): Applies to a vector field and produces a scalar field that measures the magnitude of the vector field's source or sink at a point. Example: $\nabla \cdot D$ equals the charge density $\rho_v$, indicating charges are sources of electric displacement.
• Curl ($\nabla \times$): Applies to a vector field and produces a vector field that measures the rotation or circulation of the vector field at a point. Example: $\nabla \times H$ equals the current density $J$ (plus displacement current), indicating currents are sources of magnetic fields that curl around them.

These operators allow us to express the fundamental laws of electromagnetism as differential equations, providing a localized description of how fields behave at every point in space.