Consider a cube of side $a = 0.1 m$ placed such that its six faces are given by equation $x = 0, x = + a, y = 0, y = + a, z = 0$ and $z = + a$ . If its placed in a electric field given by $\to E = x^{2}^i + y^j$ N/C, the electric flux crossing out of the cube in the unit of $10^{- 4} N m^{2} / C$ is

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Solution

Flux through any surface us given by $ϕ = \int \to E . - \to d A$

Flux through face $A B C D (x = 0, y = 0 to a) \to E = y^j - \to d A = d A (-^i) d ϕ = \to E . - \to d A = 0 (∵^j .^i = 0) \Rightarrow ϕ_{A B C D} = 0 Flux through face E F G H (x = a, y = 0 t o a) \to E = a^{2}^i + y^j - \to d h = d A^i d ϕ = \to E . - \to d A = a^{2} d A ϕ_{E F G H} = a^{4} = 10^{- 4} N m^{2} C^{- 1} Flux through face C D G H (y = 0, x = 0 t o) \to E = x^{2}^i - \to d A = d A (-^j) d ϕ = 0 (∵^i .^j = 0) ϕ_{C D G H} = 0 Flux through face A B F E (x = 0 t o a, y = a) \to E = x^{2}^i + a^j - \to d A = d A^j d ϕ = \to E . - \to d A = a d A ϕ_{A B F E} = a \int d A = a^{3} = 10^{- 3} N m^{2} C^{- 1} Flux through face A E G C and B F H D (x = 0 t o a, y = 0 t o a) \to E = x^{2}^i + y^j - \to d A = d A (\pm^k) d ϕ = \to E . - \to d A = 0 (∵^i .^k =^j .^k = 0) ϕ_{A E G C} = ϕ_{B F H D} = 0 Total flux through cube ϕ_{n e t} = ϕ_{E F G H} + ϕ_{A B F E} = 10^{- 4} + 10^{- 3} = 11 \times 10^{- 4} N m^{2} C^{- 1}$

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