Understanding Maxwell's Equations

Maxwell's equations are the fundamental laws governing electricity and magnetism. They describe how electric and magnetic fields are generated and altered by each other and by charges and currents. There are four main equations:

• Gauss's Law for Electricity: \(\nabla \cdot \mathbf{D} = \rho_f\)
• Gauss's Law for Magnetism: \(\nabla \cdot \mathbf{B} = 0\)
• Faraday's Law of Induction: \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\)
• Ampère-Maxwell Law: \(\nabla \times \mathbf{H} = \mathbf{J}_f + \frac{\partial \mathbf{D}}{\partial t}\)

In these equations:

• \(\mathbf{D}\) represents the electric displacement field.
• \(\mathbf{B}\) represents the magnetic field.
• \(\mathbf{E}\) represents the electric field.
• \(\mathbf{H}\) represents the magnetic field intensity.
• \(\rho_f\) is the free charge density, a source of electric fields.
• \(\mathbf{J}_f\) is the free current density, a source of magnetic fields.

An equation is classified as inhomogeneous if it includes source terms like \(\rho_f\) or \(\mathbf{J}_f\). Based on this, Gauss's Law for Electricity and the Ampère-Maxwell Law are the two inhomogeneous Maxwell's equations.

Properties of a Good Conductor

A material is considered a good conductor when its electrical conductivity (\(\sigma\)) is very high (approaching infinity, \(\sigma \to \infty\)). This property significantly simplifies the behavior of electric and magnetic fields within the conductor:

• Electric Field: The electric field inside a good conductor is practically zero (\(\mathbf{E} \approx 0\)). This is because charges move freely to cancel out any internal field.
• Free Charge Density: Since \(\mathbf{E} \approx 0\) inside, the free charge density (\(\rho_f\)) within the bulk of the conductor must also be zero (\(\rho_f = 0\)). Any net free charge resides on the surface of the conductor.
• Free Current Density: Ohm's Law relates current density to the electric field: \(\mathbf{J}_f = \sigma \mathbf{E}\). In a good conductor, even with a small \(\mathbf{E}\) (which is close to zero), the product \(\sigma \mathbf{E}\) can be significant. However, because \(\mathbf{E} \approx 0\) inside the conductor, the free current density also becomes negligible (\(\mathbf{J}_f \approx 0\)).
• Electric Displacement Field: The electric displacement field is defined as \(\mathbf{D} = \epsilon \mathbf{E}\), where \(\epsilon\) is the permittivity. Since \(\mathbf{E} \approx 0\) inside the conductor, \(\mathbf{D}\) is also approximately zero (\(\mathbf{D} \approx 0\)).

Maxwell's Equations Within a Good Conductor

Now, let's see how these conditions affect Maxwell's equations specifically *inside* the material of a good conductor:

1. Gauss's Law for Electricity: \(\nabla \cdot \mathbf{D} = \rho_f\)

   In a good conductor, we have \(\rho_f = 0\) and \(\mathbf{D} \approx 0\). Substituting these values, the equation becomes \(\nabla \cdot \mathbf{D} = 0\) (or \(0 = 0\)). The source term \(\rho_f\) is absent, making this equation effectively homogeneous within the conductor.

2. Gauss's Law for Magnetism: \(\nabla \cdot \mathbf{B} = 0\)

   This equation has no source terms and is inherently homogeneous. It remains homogeneous inside the conductor.

3. Faraday's Law: \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\)

   Since \(\mathbf{E} \approx 0\) inside the conductor, the left side is zero: \(\nabla \times (0) = 0\). This implies \(0 = -\frac{\partial \mathbf{B}}{\partial t}\), meaning the magnetic field does not change with time (\(\frac{\partial \mathbf{B}}{\partial t} = 0\)). This inherently homogeneous equation remains homogeneous.

4. Ampère-Maxwell Law: \(\nabla \times \mathbf{H} = \mathbf{J}_f + \frac{\partial \mathbf{D}}{\partial t}\)

   In a good conductor, both \(\mathbf{J}_f \approx 0\) and \(\mathbf{D} \approx 0\) (implying \(\frac{\partial \mathbf{D}}{\partial t} \approx 0\)). The equation simplifies to \(\nabla \times \mathbf{H} = 0 + 0\), or \(\nabla \times \mathbf{H} = 0\). The source terms \(\mathbf{J}_f\) and \(\frac{\partial \mathbf{D}}{\partial t}\) are absent, making this equation also homogeneous within the conductor.

Conclusion

While Maxwell's equations include two inhomogeneous equations in general (Gauss's law for electricity and the Ampère-Maxwell law), the specific conditions within a good conductor cause the source terms (\(\rho_f\) and \(\mathbf{J}_f\)) in these equations to become zero inside the material. As a result, both equations behave as homogeneous equations within the conductor.

Therefore, the number of inhomogeneous Maxwell's equations applicable to the fields *inside* a good conductor is zero.