Gauss's Law

The electric flux through a surface is the field component perpendicular to it, summed over the whole surface: Φ_E = ∫ E⋅dA, a signed scalar measured in N m² C⁻¹ that counts how much field crosses. Use the supplied topic model Y_out = s*s, where Y_out is PANEL AREA A (m²). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. s is Panel side, measured in m. Keep t (Angle θ from normal to field) = 55 °; ef (Field magnitude E) = 300 N C⁻¹ fixed. For this exercise use only 0.2425 m ≤ s ≤ 0.5575 m 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 = 0.18892062 m², with a bounded reading uncertainty ±0.018283 m². Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown s by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0000315 m, 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.

Gauss's law says the net electric flux out of any closed surface equals the net charge enclosed by that surface divided by ε₀, whatever the surface's shape and whatever charges sit outside it. Use the supplied topic model Y_out = 89.88*(qin/(rad_*rad_)+qout/((11+rad_)*(11+rad_))), where Y_out is FAR-SIDE E_r (kN C⁻¹). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. rad_ is Gaussian radius, measured in cm. Keep qin (Enclosed charge q_in) = 3 nC; qout (Outside charge q_out) = 4 nC fixed. For this exercise use only 2.925 cm ≤ rad_ ≤ 6.075 cm 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 = 12.911344 kN C⁻¹, with a bounded reading uncertainty ±0.6558 kN C⁻¹. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown rad_ by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000315 cm, 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.

A Gaussian surface is an imaginary closed surface chosen so that symmetry makes ∮E⋅dA trivial: on every piece, either E is constant in magnitude and perpendicular to the surface, or the flux through that piece is zero. Use the supplied topic model Y_out = 898.8*q - 89880*p/(d-10)^2, where Y_out is E⊥ NEAR SIDE (N C⁻¹). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. d is q₂ distance from centre, measured in cm. Keep q (Enclosed charge q₁) = 3 nC; p (Outside charge q₂) = 2 nC fixed. For this exercise use only 20.8687 cm ≤ d ≤ 42.1312 cm 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 = 2380.0837 N C⁻¹, with a bounded reading uncertainty ±21.33 N C⁻¹. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown d by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00212625 cm, 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.

A spherically symmetric charge distribution is one whose charge density depends only on the distance r from a single centre; its field is radial everywhere, and at radius r it is fixed entirely by the charge enclosed within r. Use the supplied topic model Y_out = (539.28/rs^2)*clamp(((rp/rs)^3-c^3)/(1-c^3),0,1)/(rp/rs)^2, where Y_out is FIELD AT PROBE (kV m⁻¹). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; clamp(x,a,b) confines x to [a,b]. rp is Probe radius r, measured in cm. Keep rs (Outer radius R) = 6 cm; c (Empty core fraction a/R) = 0 fixed. For this exercise use only 6.66687 cm ≤ rp ≤ 17.9281 cm 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 = 2.9431906 kV m⁻¹, with a bounded reading uncertainty ±0.18981 kV m⁻¹. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown rp by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00112612 cm, 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.