University Physics II · topic UPII-05.04
Dielectric polarization
Connect the ideas, make your working visible, and test the result. Use the hints when you need a starting point; open the full solution after your attempt.
Further practice
Work through a fresh case before tackling the extension.
Dielectric Polarization: predict a finite change
Polarization is the displacement or alignment of a material's bound charge under an applied field, measured by P, the dipole moment per unit volume in C m⁻². Its surfaces then carry bound charge that partly cancels the applied field. Use the supplied topic model Y_out = (sf/8.85)/(1+xe), where Y_out is FIELD IN SLAB E (MV m⁻¹). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. xe is Susceptibility χ_e, dimensionless. Keep sf (Free charge density σ_f) = 8.85 μC m⁻² fixed. For this exercise use only 0.95625 ≤ xe ≤ 8.04375 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 xe = 2.5155 and 3.50775. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δxe across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
FIELD IN SLAB E against xe
Predictions from the supplied Dielectric Polarization model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): FIELD IN SLAB E / MV m⁻¹
Horizontal axis (x): xe: Susceptibility χ_e
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| xe: Susceptibility χ_e | FIELD IN SLAB E / MV m⁻¹ |
|---|---|
| 0.9562499999999999 | 0.5111821086261982 |
| 2.728125 | 0.26823134953897737 |
| 4.5 | 0.18181818181818182 |
| 6.271875 | 0.13751611516974646 |
| 8.04375 | 0.11057360055286801 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of xe separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At xe = 2.5155, substitute into Y_out = (sf/8.85)/(1+xe): Y₀ = 0.284455 MV m⁻¹.
- At xe = 3.50775, the same fixed-parameter model gives Y₁ = 0.22184 MV m⁻¹.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.0626144 MV m⁻¹; the input change is Δxe = 0.99225.
- Average sensitivity is ΔY_out/Δxe = -0.0631034 MV m⁻¹ per unit input. Its negative sign means that this response decreases over the specified interval.
- This is a secant slope between two settings. It equals a point derivative only for a locally linear relation or in a justified small-interval limit.
Answer Y₀ = 0.284455 MV m⁻¹, Y₁ = 0.22184 MV m⁻¹; ΔY_out = -0.0626144 MV m⁻¹; average sensitivity = -0.0631034 MV m⁻¹ per unit input.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Extra challenging questions
Work through the linked stages and explain what each result means.
Dielectric Polarization: infer an input with uncertainty
Polarization is the displacement or alignment of a material's bound charge under an applied field, measured by P, the dipole moment per unit volume in C m⁻². Its surfaces then carry bound charge that partly cancels the applied field. Use the supplied topic model Y_out = (sf/8.85)/(1+xe), where Y_out is FIELD IN SLAB E (MV m⁻¹). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. xe is Susceptibility χ_e, dimensionless. Keep sf (Free charge density σ_f) = 8.85 μC m⁻² fixed. For this exercise use only 0.95625 ≤ xe ≤ 8.04375 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 = 0.15924518 MV m⁻¹, with a bounded reading uncertainty ±0.0073007 MV 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 xe by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00070875, 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.
FIELD IN SLAB E against xe
Predictions from the supplied Dielectric Polarization model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): FIELD IN SLAB E / MV m⁻¹
Horizontal axis (x): xe: Susceptibility χ_e
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| xe: Susceptibility χ_e | FIELD IN SLAB E / MV m⁻¹ |
|---|---|
| 0.9562499999999999 | 0.5111821086261982 |
| 2.728125 | 0.26823134953897737 |
| 4.5 | 0.18181818181818182 |
| 6.271875 | 0.13751611516974646 |
| 8.04375 | 0.11057360055286801 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(xe) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(0.95625) = 0.511182 MV m⁻¹ and Y_out(8.04375) = 0.110574 MV m⁻¹. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives xe_line = 0.95625 + (0.159245 − 0.511182)(8.04375 − 0.95625)/(0.110574 − 0.511182) = 7.18266. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives xe = 5.27962; substituting back gives Y_out = 0.15924518 MV m⁻¹.
- The interpolation error in the input is |xe_line − xe| = 1.90304. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.15194448 MV m⁻¹ and 0.16654588 MV m⁻¹. Sort the roots even if the response decreases: 5.00435 ≤ xe ≤ 5.58135.
- The lower and upper input deviations are 0.275273 and 0.301726. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer xe ≈ 5.27962; compatible interval [5.00435, 5.58135]. Endpoint interpolation gives 7.18266.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.