First slide

Electric field

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 $λ$ .

Moderate

A

$\frac{λ}{3 π ε_{0} a} (\hat{i} + \hat{j} + \hat{k})$
B

$\frac{λ}{2 π ε_{0} a} (\hat{i} + \hat{j} + \hat{k})$
C

$\frac{λ}{2 \sqrt{2} π ε_{0} a} (\hat{i} + \hat{j} + \hat{k})$
D

$\frac{\sqrt{2} λ}{π ε_{0} a} (\hat{i} + \hat{j} + \hat{k})$

Solution

Let us consider the electric field due to wire (3) only.
$\begin{array}{l} {\vec{E}}_{3} = E \hat{u} \\ {\vec{E}}_{3} = \frac{λ}{2 π ε_{0} {(a^{2} + a^{2})}^{1 / 2}} (\hat{i} \cos 45^{\circ} + \hat{j} \sin 45^{\circ}) \\ = \frac{λ}{2 \sqrt{2} π ε_{0} a} \frac{1}{\sqrt{2}} (\hat{i} + \hat{j}) \\ {\vec{E}}_{3} = \frac{λ}{4 π ε_{0} a} (\hat{i} + \hat{j}) \end{array}$
Similarly, electric field due to wires (1) and (2)
$\begin{array}{l} {\vec{E}}_{1} = \frac{λ}{4 π ε_{0} a} (\hat{j} + \hat{k}) and {\vec{E}}_{2} = \frac{λ}{4 π ε_{0} a} (\hat{i} + \hat{k}) \\ {\vec{E}}_{net} = {\vec{E}}_{1} + {\vec{E}}_{2} + {\vec{E}}_{3} {\vec{E}}_{n e t} = \frac{λ}{2 π ε_{0} a} (\hat{i} + \hat{j} + \hat{k}) \end{array}$

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