An infinitely long wire has uniform linear charge density λ = 2 nC/m. It passes through two maximally separated corners of a Gaussian cube of side √3 cm. Neglect edge effects. If the net electric flux is x N m² C⁻¹ and 1/(4πε₀) = 9×10⁹ SI units, then x is:
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Question

An infinitely long wire has uniform linear charge density λ = 2 nC/m. It passes through two maximally separated corners of a Gaussian cube of side √3 cm. Neglect edge effects. If the net electric flux is x N m² C⁻¹ and 1/(4πε₀) = 9×10⁹ SI units, then x is:

Options

G1 Physics · Accepted Answer

The maximally separated corners of a cube are opposite vertices, so the length of wire inside the cube is the body diagonal. If cube side is a = √3 cm, body diagonal = √3a = 3 cm = 3×10⁻² m. Enclosed charge: q e n c q_enc q e ​ n c = λ(3×10⁻²) = (2×10⁻⁹)(3×10⁻²) = 6×10⁻¹¹ C. By Gauss's law, Φ = q e n c / ε 0 . q_enc/ε₀. q e ​ n c / ε 0 ​ . Given 1/(4πε₀) = 9×10⁹ ⇒ 1/ε₀ = 36π×10⁹. Thus, Φ = (6×10⁻¹¹)(36π×10⁹) = 2.16π N m² C⁻¹. Hence x = 2.16π. Correct option: D.