As shown in the figure, a cuboid lies in a region with electric field ⃗E=(2x²̂i-4ŷj+6̂k) N C⁻¹. The magnitude of the charge within the cuboid is nε₀ C. The value of n is (the dimensions of the cuboid are 1×2×3 m³, with one corner at (0,0,0) and the adjacent corners at (1,0,0), (0,2,0) and (0,0,3)).

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Accepted Answer

12. Take the faces one pair at a time; each pair of opposite faces sees only the matching component of ⃗E. x-faces (area 2×3=6): at x=1, φ=2(1)²×6=12; at x=0, Eₓ=0, so φ=0. y-faces (area 1×3=3): at y=2, φ=(-4×2)×3=-24; at y=0, E_y=0, so φ=0. z-faces: E_z=6 is constant, so the entry and exit fluxes cancel exactly. Net flux =12-24=-12 N m² C⁻¹, and q=ε₀φ=-12ε₀ C. The magnitude of the enclosed charge is 12ε₀ C, so n=12.

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