Conductors, insulators, and polarization

A conductor carries charge that is free to travel through the whole sample, an insulator holds every charge bound to its own molecule, and polarization is the small separation of bound charge that a field produces inside neutral matter. Use the supplied topic model Y_out = 0.16689*nd*ef, where Y_out is POLARIZATION P = σ_b (μC m⁻²). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. nd is Molecules per cubic metre n, measured in 10²⁷ m⁻³. Keep ef (Applied field E) = 3 MV m⁻¹ fixed. For this exercise use only 13.1875 10²⁷ m⁻³ ≤ nd ≤ 36.8125 10²⁷ m⁻³ on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. Calculate Y_out at nd = 18.385 10²⁷ m⁻³ and 21.6925 10²⁷ m⁻³. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δnd across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.

A conductor carries charge that is free to travel through the whole sample, an insulator holds every charge bound to its own molecule, and polarization is the small separation of bound charge that a field produces inside neutral matter. Use the supplied topic model Y_out = 0.16689*nd*ef, where Y_out is POLARIZATION P = σ_b (μC m⁻²). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. nd is Molecules per cubic metre n, measured in 10²⁷ m⁻³. Keep ef (Applied field E) = 3 MV m⁻¹ fixed. For this exercise use only 13.1875 10²⁷ m⁻³ ≤ nd ≤ 36.8125 10²⁷ m⁻³ on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 13.817866 μC m⁻², with a bounded reading uncertainty ±0.69196 μC m⁻². Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown nd by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0023625 10²⁷ m⁻³, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.