Q: Consider an electric field $\displaystyle \vec{E} = E_0 \hat{x}$ ; where E_{0} is a constant. The flux through the shaded area (as shown in the Fig) due to this filed is

(a) 2 E_{0} a^{2}

(b) √2 E_{0} a^{2}

(c) E_{0} a^{2}

(d) (E_{0} a^{2})/√2

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Sol: Let $ \hat{x} , \hat{y} , \hat{z}$ be the unit vectors along X-axis , Y -axis & Z -axis respectively .

Area vector $\displaystyle \vec{A} = \vec{PQ} \times \vec{PS} $

$\displaystyle \vec{A} = (0 \hat{x} + a \hat{y} + 0 \hat{z}) \times (a \hat{x} + 0 \hat{y} + a \hat{z}) $

$\displaystyle \vec{A} = (a^2 \hat{x}-a^2 \hat{z}) $

Electric Flux through shaded Area ;

$\displaystyle \phi = \vec{E}.\vec{A} =( E_0\hat{x}).(a^2 \hat{x}-a^2 \hat{z}) $

$\displaystyle \phi = E_0 a^2 $