Derivation of Maxwell's stress and electrostatic attraction using Maxwell's equations

Problem Statement

Two perfectly conductive plates of sufficiently large area are placed parallel to each other at a distance $d$d in a vacuum. A DC power supply is connected between the two conductive plates, providing a constant potential difference $V$V between the plates. The space between the plates is filled with vacuum (permittance $\varepsilon_0$ε0​).

Let $S$S be the area of ​​the conductive plates, and assume that all edge effects are negligible. Also, assume that the electric field vector $\boldsymbol{E} = (E_x, E_y, E_z)$E=(Ex​,Ey​,Ez​) between the plates is uniform and points in the direction normal to the plates (in the $x$x axis direction).

From Maxwell's equations and general theory of electromagnetism, each component $T_{ij}$Tij​ ($i, j \in \{x, y, z\}$i,j∈{x,y,z}) of Maxwell's stress tensor $\boldsymbol{T}$T in a vacuum is defined using the electric field vector as follows:

$T_{ij} = \varepsilon_0 \left( E_i E_j - \frac{1}{2} \delta_{ij} |\boldsymbol{E}|^2 \right)$Tij​=ε0​(Ei​Ej​−21​δij​∣E∣2) Here, $\delta_{ij}$δij​ represents the Kronecker delta.

The magnitude of the electrostatic attractive force $F$F exerted by one conducting plate on another in the $x$x axis can be calculated exactly by integrating the normal component of Maxwell's stress tensor over the closed surface $A$A surrounding the plates, as follows: $F = \oint_A \sum_{j} T_{xj} n_j dR$F=∮A​∑j​Txj​nj​dR Here, $\boldsymbol{n} = (n_x, n_y, n_z)$n=(nx​,ny​,nz​) is the outward unit normal vector of the closed surface $A$A.

Using this tensor calculation, express the normal component $T_{xx}$Txx​ in terms of $\varepsilon_0, V, and d$ε0​,V,andd, and derive an algebraic expression $F$F representing the magnitude of the total electrostatic attractive force acting on one of the plates.

Constraints

• $\varepsilon_0 = 8.85 \times 10^{-12} \text{ F/m}$ε0​=8.85×10−12 F/m
• $S = 0.500 \text{ m}^2$S=0.500 m2
• $V = 3000 \text{ V}$V=3000 V
• $d = 2.00 \times 10^{-3} \text{ m}$d=2.00×10−3 m

Input Format

Calculate the magnitude of the electrostatic attractive force $F \text{ [N]}$F [N] derived to $A \times 10^{-6} \text{ N}$A×10−6 N. Find the value of the coefficient $A$A in this case, and input a natural number representing that value.