(iii) The total electric field at \( (0,0,0) \) is: \( \begin{array}{c}\overrightarrow{\mathbf{E}}_{\text {total }}=\overrightarrow{\mathbf{E}}_{1}+\overrightarrow{\mathbf{E}}_{2}=\frac{-Q_{1}}{4 \pi \varepsilon_{0} a^{2}} \overrightarrow{\mathbf{a}}_{x}+\frac{Q_{2}}{4 \pi \varepsilon_{0} a\left(a^{2}+L^{2}\right)^{\frac{1}{2}}} \overrightarrow{\mathbf{a}}_{x} \\ \text { For } \overrightarrow{\mathbf{E}}_{\text {total }}=0 \\ \frac{-Q_{1}}{\boldsymbol{a}}+\frac{Q_{2}}{\left(\boldsymbol{a}^{2}+{\left.L^{2}\right)^{\frac{1}{2}}}_{a}^{a}\right.}=\mathbf{0} \\ \therefore \boldsymbol{Q}_{1}=\frac{\left.a^{2}+L^{2}\right)^{\frac{1}{2}}}{\left(Q_{2}\right.}\end{array} \)

Did you know that the concept of the electric field dates back to the early 19th century? It was Michael Faraday and James Clerk Maxwell who laid down the foundational principles that allowed us to understand electric fields and electromagnetic theory. Their work has helped shape not only physics but also engineering, enabling countless modern technologies, from electric motors to telecommunications! In practical applications, understanding electric fields is crucial in designing circuits and electronic devices. Engineers must consider electric fields when determining the layout of circuit components to prevent interference and ensure efficient operation. It also plays a vital role in technologies like capacitors, which store energy, and in the design of various sensors used in everyday devices like smartphones and automotive systems. So, next time you use a gadget, remember the electric fields working behind the scenes!