Two parallel circular plates of radius \(a\) are located at \(z=0\) and \(z=d\). The top plate \((z=d)\) is raised to potential \(V_{0} ;\) the bottom plate is grounded. Between the plates is a conducting material having radial-dependent conductivity, \(\sigma(\rho)=\sigma_{0} \rho\), where \(\sigma_{0}\) is a constant. \((a)\) Find the \(\rho\) -independent electric field strength, \(\mathbf{E}\), between plates. \((b)\) Find the current density, \(\mathbf{J}\) between plates. ( \(c\) ) Find the total current, \(I\), in the structure. \((d)\) Find the resistance between plates.

Understanding Gauss's Law is crucial when analyzing the behavior of electric fields in various geometrical configurations. It is one of the four Maxwell's equations, governing the relation between electric charges and electric fields. Gauss's Law can be mathematically stated as \[ \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \] where \( \mathbf{E} \) is the electric field, \( d\mathbf{A} \) is a differential area vector on a closed surface, \( Q_{\text{enc}} \) is the enclosed charge within the surface, and \( \epsilon_0 \) is the vacuum permittivity.

In the provided exercise, we use Gauss's Law to conclude that the electric field strength between parallel plates is uniform and directed along the axis perpendicular to the plates. This deduction comes from the symmetry of the setup and the assumption that there is no charge within the conductive material between the plates, simplifying our calculations and leading to the expression \( E_z = \frac{V_0}{d} \) for the electric field strength.

Conductivity, denoted by \( \sigma \), is a material property that quantifies how well a substance can conduct electric current. It is the inverse of resistivity and directly affects the current density within a material when exposed to an electric field. In the case of radial-dependent conductivity, \( \sigma(\rho) = \sigma_0 \rho \), we see that conductivity is not constant across the material but instead varies with the radial distance from the center.

The variable conductivity within the conducting material sandwiched between parallel plates affects the current density throughout the span of the material. Such a spatially varying conductivity may arise in composite materials or those with non-uniform doping, challenging us to integrate current density over the entire area to calculate the total current.

Current density, represented by \( \mathbf{J} \), is a vector quantity that describes the amount of electric current flowing per unit area of a cross-section. The relationship between current density, conductivity, and electric field strength is given by Ohm's Law: \[ \mathbf{J} = \sigma \mathbf{E} \] In the exercise we're considering, the current density depends on radial position and is given by \( \mathbf{J} = \sigma_0 \rho E_z \hat{z} \).

To compute the total current in the structure, we integrate the current density across the area of the plate. This process involves integrating over the plane in cylindrical coordinates, taking into account the variation of current density with radius due to the radial dependence of conductivity. The resulting total current is pivotal in determining the resistance between the plates and characterizing the electrical properties of the system.

Ohm's Law is a foundational principle in electrical engineering and physics. It states that the current (\( I \)) through a conductor between two points is directly proportional to the voltage (\( V \)) across the two points, and inversely proportional to the resistance (\( R \)) of the conductor: \[ V = IR \] This relationship allows us to calculate any one of the three variables if the other two are known. However, Ohm’s Law applies to ideal conductors and may require modifications when dealing with materials with spatially dependent properties, such as the radial-dependent conductivity in our example.

In the exercise, we first calculated the current density using Ohm's Law and then integrated it to find the total current through the structure. Subsequently, we used the inverse relationship given by Ohm's Law to find the electric resistance between the plates, revealing the inherent properties of the system concerning the flow of electric charge.

Electric resistance is a measure of the opposition that a material presents to the flow of electric current. It is directly related to the conductive properties of the material, and in simple cases, inversely proportional to the material's cross-sectional area and directly proportional to length. Resistance can be calculated using resistance law: \[ R = \frac{\rho L}{A} \] where \( \rho \) is the resistivity, \( L \) is the length of the conductor, and \( A \) is the cross-sectional area.

In our case, the resistance is obtained from Ohm's Law once the total current is known. The final expression, \( R = \frac{d}{\sigma_0 \pi a^3} \), indicates that resistance in this scenario is due not only to the inherent electric properties of the material but also to its geometry and the distribution of conductivity. The dependence on the third power of the radius signifies that for larger plates or those with greater area, even a small increase in dimensions will result in a substantial reduction in resistance, drastically affecting the current that can be sustained by the system.