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 {{\varepsilon }_{0}}b}(\hat{i}+\hat{j}+\hat{k}) $
• B. $ \frac{\sigma }{2\pi {{\varepsilon }_{0}}b}(\hat{i}+\hat{j}+\hat{k}) $
• C. $ \frac{\sigma }{\pi {{\varepsilon }_{0}}b}(\hat{i}+\hat{j}-\hat{k}) $
• D. $ \frac{\sigma }{2\pi {{\varepsilon }_{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 {{\varepsilon }_{0}}{{({{b}^{2}}+{{b}^{2}})}^{1/2}}}(\hat{i}\cos {{45}^{o}}+\hat{k}\cos {{45}^{o}}) $
$ =\frac{\sigma }{2\sqrt{2}\pi {{\varepsilon }_{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 {{\varepsilon }_{0}}b}\times \frac{1}{\sqrt{2}}(\hat{j}\,+\hat{k}) $
and $ {{E}_{2}}=\frac{\sigma }{2\sqrt{2}\pi {{\varepsilon }_{0}}b}\times \frac{1}{\sqrt{2}}(\hat{i}\,+\hat{k}) $
$ {{E}_{net}}={{E}_{1}}+{{E}_{2}}+{{E}_{3}}=\frac{\sigma }{2\pi {{\varepsilon }_{0}}b}(\hat{i}\,+\hat{j}+\hat{k}) $