A cubical volume is bounded by the surfaces . The electric field in the region is given by where . If the charge contained in the cubical volume is The value of is __________. (Take ). [2023]

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Solution

Q.
A cubical volume is bounded by the surfaces $x = 0, x = a, y = 0, y = a, z = 0, z = a$ . The electric field in the region is given by $\vec{E} = E_{0} x \hat{i},$ where $E_{0} = 4 \times 10^{4} {NC}^{- 1} m^{- 1}$ . If $a = 2cm,$ the charge contained in the cubical volume is $Q \times 10^{- 14} C .$ The value of $Q$ is __________. (Take $ε_{0} = 9 \times 10^{- 12} C^{2} {/Nm}^{2}$ ). [2023]

Ans.
(288)

$\vec{E} = E_{0} x \hat{i}$

$ϕ_{net} = ϕ_{A B C D} = E_{0} a \cdot a^{2}$

$\frac{q_{en}}{ε_{0}} = E_{0} a^{3}$

$\Rightarrow q_{en} = E_{0} ε_{0} a^{3}$

$q_{en} = 4 \times 10^{4} \times 9 \times 10^{- 12} \times 8 \times 10^{- 6} = 288 \times 10^{- 14} C$

$\Rightarrow Q = 288$

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