Electrostatic Sensor Array Potential Gradients (Electrical & Computer Engineering)

In micro-electro-mechanical systems (MEMS) and semiconductor device fabrication at UNLV, engineers design micro-sensor arrays to manipulate and sort charged nanoparticles in cleanroom aerosol streams. Inside an electrostatic precipitation sensing chamber, the electric potential (voltage) distribution $V(x, y, z)$ (measured in volts, $\text{V}$) across a 3D coordinate grid (measured in centimeters, $\text{cm}$) is governed by the quadratic potential field:

A test sensor probe is positioned at coordinate location $P(1, 2, -2)$.

1. Calculate the spatial voltage gradient vector $\nabla V$ at the probe location $P(1, 2, -2)$, expressing your answer in both bracket notation $\langle a, b, c \rangle$ and Cartesian unit vector notation $a\hat{i} + b\hat{j} + c\hat{k}$.
2. Determine the maximum spatial rate of voltage change $\|\nabla V\|$ at point $P$, and state the direction of maximum voltage increase as a unit vector $\vec{u}_{\text{max}}$.
3. Physical Law & Nanoparticle Dynamics: In classical electromagnetism, the electric field vector $\vec{E}$ is defined as the negative gradient of electric potential:
   $$\vec{E} = -\nabla V$$
   Compute $\vec{E}(1, 2, -2)$. If a positively charged aerosol particle with net charge $q > 0$ enters this zone, explain the physical direction in which the electrostatic Coulomb force $\vec{F} = q\vec{E}$ will accelerate the particle, and how engineers position collection electrodes to capture it.