Question

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

• A. $\frac{\lambda }{3\pi {\epsilon }_{0}a}(\hat{i}+\hat{j}+\hat{k})$
• B. $\frac{\lambda }{2\pi {\epsilon }_{0}a}(\hat{i}+\hat{j}+\hat{k})$
• C. $\frac{\lambda }{2\sqrt{2}\pi {\epsilon }_{0}a}(\hat{i}+\hat{j}+\hat{k})$
• D. $\frac{\sqrt{2}\lambda }{\pi {\epsilon }_{0}a}(\hat{i}+\hat{j}+\hat{k})$

Answer 1

The correct option is B $\frac{\lambda }{2\pi {\epsilon }_{0}a}(\hat{i}+\hat{j}+\hat{k})$

Electric field due to an infinitely long wire at a distance r is given as $\frac{\lambda }{2\pi {ϵ}_{o}r}$
Let us consider the electric field due to the wire along the z axis at point p.
${\overrightarrow{E}}_z=E\hat{r}$
$=\frac{\lambda }{2\pi {\epsilon }_0(x^2+y^2{)}^{1/2}}\hat{r}=\frac{\lambda }{2\pi {\epsilon }_0(a^2+a^2{)}^{1/2}}(\hat{i}cos{45}^o+\hat{j}cos{45}^o)$
$=\frac{\lambda }{2\sqrt{2}\pi {\epsilon }_{0}a}\frac{1}{\sqrt{2}}(\hat{i}+\hat{j})$
$=\frac{\lambda }{4\pi {\epsilon }_{0}a}(\hat{i}+\hat{j})$
Similarly, electric field due to wires along x and y axes is given as
${\overrightarrow{E}}_y=\frac{\lambda }{4\pi {\epsilon }_{0}a}(\hat{i}+\hat{k})$ and ${\overrightarrow{E}}_x=\frac{\lambda }{4\pi {\epsilon }_{0}a}(\hat{j}+\hat{k})$
${\overrightarrow{E}}_{net}={\overrightarrow{E}}_x+{\overrightarrow{E}}_y+{\overrightarrow{E}}_z=\frac{\lambda }{2\pi {\epsilon }_{0}a}(\hat{i}+\hat{j}+\hat{k})$