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Problem Solving

The cube as s...

Question

The cube as shown in Fig. has sides of length $L = 10.0 c m$ . The electric field is uniform, has a magnitude $E = 4.00 \times 10^{3} N C^{- 1}$ , and is parallel to the $x y - p l a n e$ at an angle of $37^{\circ}$ measured from the $+ x - a x i s$ towards the $+ y - a x i s$ .
The total net electric flux through all faces of the cube is

8 N m^{2} C^{- 1}

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- 8 N m^{2} C^{- 1}

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24 N m^{2} C^{- 1}

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Zero

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Solution

The correct option is D Zero
Here, the electric field is given by $\to E = E cos 37^i + E sin 37^j$
Electric flux through any surface $S$ is $ϕ = \to E .^n S$
Where $^n$ is the normal outwardto the surface $S$
For surface $S_{1}$ : $^n = -^j, S_{1} = L^{2} = (0.1)^{2} m^{2}$
Thus, $ϕ_{1} = \to E .^n S_{1} = (E cos 37^i + E sin 37^j) . (- L^{2}^j) = - E L^{2} sin 37 = - (4 \times 10^{3}) (0.1)^{2} sin 37 = - 24 N m^{2} C^{- 1}$
For surface $S_{2}$ : $^n =^k, S_{2} = L^{2} = (0.1)^{2} m^{2}$
Thus, $ϕ_{2} = \to E .^n S_{2} = (E cos 37^i + E sin 37^j) . (L^{2}^k) = 0$
For surface $S_{3}$ : $^n =^j, S_{3} = L^{2} = (0.1)^{2} m^{2}$
Thus, $ϕ_{3} = \to E .^n S_{3} = (E cos 37^i + E sin 37^j) . (L^{2}^j) = E L^{2} sin 37 = (4 \times 10^{3}) (0.1)^{2} sin 37 = 24 N m^{2} C^{- 1}$
For surface $S_{4}$ : $^n = -^k, S_{4} = L^{2} = (0.1)^{2} m^{2}$
Thus, $ϕ_{4} = \to E .^n S_{4} = (E cos 37^i + E sin 37^j) . (- L^{2}^k) = 0$
For surface $S_{5}$ : $^n =^i, S_{5} = L^{2} = (0.1)^{2} m^{2}$
Thus, $ϕ_{5} = \to E .^n S_{5} = (E cos 37^i + E sin 37^j) . (L^{2}^i) = E L^{2} cos 37 = (4 \times 10^{3}) (0.1)^{2} cos 37 = 32 N m^{2} C^{- 1}$
For surface $S_{6}$ : $^n = -^i, S_{6} = L^{2} = (0.1)^{2} m^{2}$
Thus, $ϕ_{6} = \to E .^n S_{6} = (E cos 37^i + E sin 37^j) . (- L^{2}^i) = - E L^{2} cos 37 = - (4 \times 10^{3}) (0.1)^{2} cos 37 = - 32 N m^{2} C^{- 1}$
Thus, net flux $= ϕ_{1} + ϕ_{2} + ϕ_{3} + ϕ_{4} + ϕ_{5} + ϕ_{6} = - 24 + 0 + 24 + 0 + 32 - 32 = 0$

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The cube as shown in Fig. has sides of length L=10.0 cm. The electric field is uniform, has a magnitude E=4.00×103NC−1, and is parallel to the xy−plane at an angle of 37∘ measured from the +x−axis towards the +y−axis.The total net electric flux through all faces of the cube is