An electric field $\to E = 4 x^i - (y^{2} + 1)^j N/C$ passes through the box shown in figure. The flux of the electric field through surface $A B C D$ and $B C G F$ are marked as $ϕ_{1}$ and $ϕ_{11}$ respectively. The difference between ( $ϕ_{1} - ϕ_{11}$ ) is (in ${NM}^{2} / C$ )

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Solution

Flux of the electric field $\to E$ through any area $\to A$ is defined as
$ϕ = \to E \cdot \to A = E A cos θ$
Here, $θ =$ angle between electric field and area vector of a surface
For surface $A B C D$ Angle $θ = 90^{\circ}$
$∴ ϕ_{1} = \int E . A cos 90^{\circ} = 0$
For surface $B C G F ϕ_{11} = \int \to E \cdot - \to d A$
$∴ ϕ_{11} = [4 x^i - (y^{2} + 1)^j] \cdot 4^i = 16 x = 16 \times 3$
$ϕ_{11} = 48 {Nm}^{2} C$
$∴ ϕ_{1} - ϕ_{11} = - 48$

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