Electrostatics 3 Question 18
18. An infinitely long uniform line charge distribution of charge per unit length $\lambda$ lies parallel to the $y$-axis in the $y$ - $z$ plane at $z=\frac{\sqrt{3}}{2} a$ (see figure). If the magnitude of the flux of the electric field through the rectangular surface $A B C D$ lying in the $x-y$ plane with its centre at the origin is
Analytical & Descriptive Questions
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Solution:
- $A N B P$ is cross-section of a cylinder of length $L$. The line charge passes through the centre $O$ and perpendicular to paper.
$$ \begin{gathered} A M=\frac{a}{2}, M O=\frac{\sqrt{3} a}{2} \\ \therefore \quad \angle A O M=\tan ^{-1} \frac{A M}{O M} \\ =\tan ^{-1} \frac{1}{\sqrt{3}}=30^{\circ} \end{gathered} $$
Electric flux passing from the whole cylinder
$$ \varphi=\frac{q _{in}}{\varepsilon _0}=\frac{\lambda L}{\varepsilon _0} $$
$\therefore$ Electric flux passing through $A B C D$ plane surface (shown only $A B)=$ Electric flux passing through cylindrical surface $A N B$
$$ \begin{aligned} & =\frac{60^{\circ}}{360^{\circ}}(\varphi) \\ & =\frac{\lambda L}{6 \varepsilon _0} \\ \therefore \quad n & =6 \end{aligned} $$
