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 axis, respectively. The charge density i.e. charge per unit length of each wire is $\mathrm{\lambda .}$

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

Answer 1

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

Let us consider electric field due to wire 3 only.


$\stackrel{}{E_3}=E\hat{r}=\frac{\lambda }{2\pi {\epsilon }_0{\left(a^2+a^2\right)}^{\frac{1}{2}}}\left(\hat{i}\cos {45}^0+\hat{j}\cos {45}^0\right)=\frac{\lambda }{2\sqrt{2}{\mathrm{\pi \epsilon }}_{0}a}\frac{1}{\sqrt{2}}\left(\hat{i}+\hat{j}\right)=\frac{\lambda }{4{\mathrm{\pi \epsilon }}_{0}a}\left(\hat{i}+\hat{j}\right)\text{Similarly, electric field due to}\text{wires (1) and (2)}\stackrel{}{E_1}=\frac{\lambda }{4{\mathrm{\pi \epsilon }}_{0}a}\left(\hat{j}+\hat{k}\right)\mathrm{and}\stackrel{}{E_2}=\frac{\lambda }{4{\mathrm{\pi \epsilon }}_{0}a}\left(\hat{i}+\hat{k}\right)\stackrel{}{E_{\mathrm{net}}}=\stackrel{}{E_1}+\stackrel{}{E_2}+\stackrel{}{E_3}=\frac{\lambda }{2{\mathrm{\pi \epsilon }}_{0}a}\left(\hat{i}+\hat{j}+\hat{k}\right)$