An infinitely long non-conducting cylinder has uniform volume charge density ρ. A cylindrical cavity is drilled parallel to its axis. The cavity axis is displaced from the original cylinder axis by vector $\overrightarrow{a}$ , and the cavity lies completely inside the charged cylinder. The electric field everywhere inside the cavity is:
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Question

An infinitely long non-conducting cylinder has uniform volume charge density ρ. A cylindrical cavity is drilled parallel to its axis. The cavity axis is displaced from the original cylinder axis by vector $\overrightarrow{a}$ , and the cavity lies completely inside the charged cylinder. The electric field everywhere inside the cavity is:

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G1 Physics · Accepted Answer

Superpose the original cylinder of density +ρ and a cylinder of density -ρ occupying the cavity. For a point inside the cavity, E⃗ = ρr⃗/(2ε₀) - ρr⃗′/(2ε₀). Since r⃗ = a⃗ + r⃗′, E⃗ = ρa⃗/(2ε₀). Thus the field is uniform.