Field Point \( \vec{f}=z \overrightarrow{\mathbf{a}}_{z} \quad \) Source Point \( \vec{s}=x \overrightarrow{\mathbf{a}}_{x}+y \overrightarrow{\mathbf{a}}_{y} \) Source element \( d s=d x d y \) \[ \begin{array}{r}\vec{R}=\vec{f}-\vec{s}=z \vec{a}_{z}-\left(x \vec{a}_{x}+y \vec{a}_{y}\right) \\ \qquad \begin{array}{r}\overrightarrow{\mathrm{E}}=\sqrt{x^{2}+y^{2}+z^{2}} \\ \\ =\frac{1}{4 \pi \varepsilon_{0}} \int_{-a-a}^{a} \int_{-a}^{a} \frac{\rho_{0} d s}{R^{3}} \vec{R} \\ 4 \pi \varepsilon_{0} \\ \int_{-a}^{a} \int_{-a}^{a} \frac{z \vec{a}_{z}+\left(x \vec{a}_{x}+y \vec{a}_{y}\right)}{\left(x^{2}+y^{2}+z^{2}\right)^{\frac{3}{2}}} d x d y\end{array}\end{array} . \]

Did you know that the concept of electric fields, which this problem is tackling, dates back to the experiments of scientists like Michael Faraday and James Clerk Maxwell? In the 19th century, Faraday introduced the idea of electric fields as a way of explaining how electric forces act at a distance without contact. Maxwell later mathematically unified electricity and magnetism into a comprehensive set of equations—popularly known as Maxwell's equations—that describe how electric and magnetic fields interact. Isn't it fascinating how these historical breakthroughs lay the groundwork for our understanding of electromagnetism today? When diving into the realm of electric fields, one common mistake is neglecting to visualize the geometry involved. It’s crucial to draw diagrams to represent the source and field points clearly. Misinterpreting the direction of vectors or the distances involved can lead to incorrect calculations. Also, keep an eye on the units! Working with SI units (meters, coulombs, etc.) consistently will save you a headache later on. So grab that pencil, sketch it out, and watch the math come to life!