Gauss's Divergence Theorem and Multiple Integration in a Non-Uniform Electric Field

Problem Statement

A non-uniform electric field $\vec{E}$E exists in vacuum and is expressed in Cartesian coordinates $(x, y, z)$(x,y,z) as:

$$\vec{E} = (k x y^2) \hat{i} + (k x^2 y) \hat{j} + (\gamma z^3) \hat{k}$$E=(kxy2)i^+(kx2y)j^​+(γz3)k^

where $\hat{i}, \hat{j}, \hat{k}$i^,j^​,k^ are unit vectors along each axis, and $k$k and $\gamma$γ are constants.

Let the region $V$V be the cylinder $x^2 + y^2 \le R^2$x2+y2≤R2, $0 \le z \le H$0≤z≤H.

Find the total charge $Q$Q contained in region $V$V.

Constraints

• Constant: $k = 2\text{ V/m}^4$k=2 V/m4
• Constant: $\gamma = 1\text{ V/m}^4$γ=1 V/m4
• Cylinder radius: $R = 3\text{ m}$R=3 m
• Cylinder height: $H = 4\text{ m}$H=4 m
• Permittivity of vacuum: $\varepsilon_0 = \frac{1}{36\pi \times 10^9}\text{ F/m}$ε0​=36π×1091​ F/m

Input Format

Find the total charge $Q$Q in $\text{nC}$nC (nanocoulombs, $1\text{ nC} = 10^{-9}\text{ C}$1 nC=10−9 C) and give the answer as a positive integer.