Find the electric field vector at $P(a, a, a)$ due to three infinitely long lines of charges along $x$-, $y$- and $z$- axes respectively. The charge density, i.e., charge per unit length of each wire is $\lambda$.

Question

Aicharya · Accepted Answer

Let us consider the electric field due to wire (3) only.

$\begin{array}{l}
\overrightarrow{E_{3}}=E \hat{u} \\
\vec{E}_{3}=\frac{\lambda}{2 \pi \varepsilon_{0}\left(a^{2}+a^{2}\right)^{1 / 2}}\left(\hat{i} \cos 45^{\circ}+\hat{j} \cos 45^{\circ}\right) \\
=\frac{\lambda}{2 \sqrt{2} \pi \varepsilon_{o} a} \frac{1}{\sqrt{2}}(\hat{i}+\hat{j}) \\
\quad \vec{E}_{3}=\frac{\lambda}{4 \pi \varepsilon_{o} a}(\hat{i}+\hat{j})
\end{array}$
Similarly, electric field due to wires (1) and (2)
$\begin{array}{l}
\vec{E}_{1}=\frac{\lambda}{4 \pi \varepsilon_{0} a}(\hat{j}+\hat{k})_{\text {and }} \quad \overrightarrow{E_{2}}=\frac{\lambda}{4 \pi \varepsilon_{0} a}(\hat{i}+\hat{k}) \\
\vec{E}_{\text {net }}=\overrightarrow{E_{1}}+\overrightarrow{E_{2}}+\overrightarrow{E_{3}} \\
\vec{E}_{\text {net }}=\frac{\lambda}{2 \pi \varepsilon_{0} a}(\hat{i}+\hat{j}+\hat{k})
\end{array}$

Find the electric field vector at \(P(a, a, a)\) due to three infinitely long lines of charges along \(x\)-,…

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