Question

Find the electric field vector at $P(b,b,b)$ 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 $\sigma .$

• A. $\frac{\sigma }{\pi {\epsilon }_{0}b}(\hat{i}+\hat{j}+\hat{k})$
• B. $\frac{\sigma }{2\pi {\epsilon }_{0}b}(\hat{i}+\hat{j}+\hat{k})$
• C. $\frac{\sigma }{\pi {\epsilon }_{0}b}(\hat{i}+\hat{j}-\hat{k})$
• D. $\frac{\sigma }{2\pi {\epsilon }_{0}b}(\hat{i}+\hat{j}-\hat{k})$

Answer 1

Answer: (B)

Electric field at P. due to wire 3 only is
$E_3=\frac{\sigma }{2\pi {\epsilon }_0{(b^2+b^2)}^{1/2}}(\hat{i}\cos {45}^o+\hat{k}\cos {45}^o)$
$=\frac{\sigma }{2\sqrt{2}\pi {\epsilon }_{0}b}\times \frac{1}{\sqrt{2}}(\hat{i}+\hat{k})$
Similarly, electric field due to wires 1 and 2
$E_1=\frac{\sigma }{2\sqrt{2}\pi {\epsilon }_{0}b}\times \frac{1}{\sqrt{2}}(\hat{j}+\hat{k})$
and $E_2=\frac{\sigma }{2\sqrt{2}\pi {\epsilon }_{0}b}\times \frac{1}{\sqrt{2}}(\hat{i}+\hat{k})$
$E_{net}=E_1+E_2+E_3=\frac{\sigma }{2\pi {\epsilon }_{0}b}(\hat{i}+\hat{j}+\hat{k})$