Compare path integrals with endpoint energy
Integrate field components along each displacement and test path independence.
Learning outcome: Change B through −2, 0 and +2 while holding A fixed. Predict segment integrals, compare both cumulative curves, and independently integrate the straight path. Reverse the path and explain which signs reverse. Do not interpret the displayed segment parameter as physical distance or travel time.
Starting recap
Keep source normalization, potential reference, signed energy and evidence independence explicit.
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Try
For A = 2 and B = 1, delta V = 13 V. Horizontal-first work per charge is −8 − 5 = −13 J/C; vertical-first gives −3 − 10 = −13 J/C. For q = −2 nC, field work is +26 nJ and delta U = −26 nJ.
Open the experimentPredict
Two different paths connect the same endpoints in this static electrostatic field. Which quantity must agree?
Experiment
Change B through −2, 0 and +2 while holding A fixed. Predict segment integrals, compare both cumulative curves, and independently integrate the straight path. Reverse the path and explain which signs reverse. Do not interpret the displayed segment parameter as physical distance or travel time.
Compare two paths through one electrostatic field
Worked teaching model · synthetic values · V = Ax² + Bxy + 3y² in SI units. Static field is negative spatial gradient. Compare paths from (0,0) to (2,1) m, one horizontal then vertical and one vertical then horizontal, plus straight path. Work per positive unit charge is integral E dot dr.
Reset keeps your prediction, captures and practice history.
Cumulative horizontal-first integral
Axes rescale independently. Curves join sampled coordinates; a joining segment is not additional measured evidence. Read physical units rather than screen angles or apparent slopes.
Cumulative horizontal-first integral: numerical coordinates
| path parameter 0..2 (not distance) | integral E dot dr (J/C) |
|---|---|
| 0 | 0 |
| 0.0333333 | -0.00888889 |
| 0.0666667 | -0.0355556 |
| 0.1 | -0.08 |
| 0.133333 | -0.142222 |
| 0.166667 | -0.222222 |
| 0.2 | -0.32 |
| 0.233333 | -0.435556 |
| 0.266667 | -0.568889 |
| 0.3 | -0.72 |
| 0.333333 | -0.888889 |
| 0.366667 | -1.07556 |
| 0.4 | -1.28 |
| 0.433333 | -1.50222 |
| 0.466667 | -1.74222 |
| 0.5 | -2 |
| 0.533333 | -2.27556 |
| 0.566667 | -2.56889 |
| 0.6 | -2.88 |
| 0.633333 | -3.20889 |
| 0.666667 | -3.55556 |
| 0.7 | -3.92 |
| 0.733333 | -4.30222 |
| 0.766667 | -4.70222 |
| 0.8 | -5.12 |
| 0.833333 | -5.55556 |
| 0.866667 | -6.00889 |
| 0.9 | -6.48 |
| 0.933333 | -6.96889 |
| 0.966667 | -7.47556 |
| 1 | -8 |
| 1.03333 | -8.07 |
| 1.06667 | -8.14667 |
| 1.1 | -8.23 |
| 1.13333 | -8.32 |
| 1.16667 | -8.41667 |
| 1.2 | -8.52 |
| 1.23333 | -8.63 |
| 1.26667 | -8.74667 |
| 1.3 | -8.87 |
| 1.33333 | -9 |
| 1.36667 | -9.13667 |
| 1.4 | -9.28 |
| 1.43333 | -9.43 |
| 1.46667 | -9.58667 |
| 1.5 | -9.75 |
| 1.53333 | -9.92 |
| 1.56667 | -10.0967 |
| 1.6 | -10.28 |
| 1.63333 | -10.47 |
| 1.66667 | -10.6667 |
| 1.7 | -10.87 |
| 1.73333 | -11.08 |
| 1.76667 | -11.2967 |
| 1.8 | -11.52 |
| 1.83333 | -11.75 |
| 1.86667 | -11.9867 |
| 1.9 | -12.23 |
| 1.93333 | -12.48 |
| 1.96667 | -12.7367 |
| 2 | -13 |
Cumulative vertical-first integral
Axes rescale independently. Curves join sampled coordinates; a joining segment is not additional measured evidence. Read physical units rather than screen angles or apparent slopes.
Cumulative vertical-first integral: numerical coordinates
| path parameter 0..2 (not distance) | integral E dot dr (J/C) |
|---|---|
| 0 | 0 |
| 0.0333333 | -0.00333333 |
| 0.0666667 | -0.0133333 |
| 0.1 | -0.03 |
| 0.133333 | -0.0533333 |
| 0.166667 | -0.0833333 |
| 0.2 | -0.12 |
| 0.233333 | -0.163333 |
| 0.266667 | -0.213333 |
| 0.3 | -0.27 |
| 0.333333 | -0.333333 |
| 0.366667 | -0.403333 |
| 0.4 | -0.48 |
| 0.433333 | -0.563333 |
| 0.466667 | -0.653333 |
| 0.5 | -0.75 |
| 0.533333 | -0.853333 |
| 0.566667 | -0.963333 |
| 0.6 | -1.08 |
| 0.633333 | -1.20333 |
| 0.666667 | -1.33333 |
| 0.7 | -1.47 |
| 0.733333 | -1.61333 |
| 0.766667 | -1.76333 |
| 0.8 | -1.92 |
| 0.833333 | -2.08333 |
| 0.866667 | -2.25333 |
| 0.9 | -2.43 |
| 0.933333 | -2.61333 |
| 0.966667 | -2.80333 |
| 1 | -3 |
| 1.03333 | -3.07556 |
| 1.06667 | -3.16889 |
| 1.1 | -3.28 |
| 1.13333 | -3.40889 |
| 1.16667 | -3.55556 |
| 1.2 | -3.72 |
| 1.23333 | -3.90222 |
| 1.26667 | -4.10222 |
| 1.3 | -4.32 |
| 1.33333 | -4.55556 |
| 1.36667 | -4.80889 |
| 1.4 | -5.08 |
| 1.43333 | -5.36889 |
| 1.46667 | -5.67556 |
| 1.5 | -6 |
| 1.53333 | -6.34222 |
| 1.56667 | -6.70222 |
| 1.6 | -7.08 |
| 1.63333 | -7.47556 |
| 1.66667 | -7.88889 |
| 1.7 | -8.32 |
| 1.73333 | -8.76889 |
| 1.76667 | -9.23556 |
| 1.8 | -9.72 |
| 1.83333 | -10.2222 |
| 1.86667 | -10.7422 |
| 1.9 | -11.28 |
| 1.93333 | -11.8356 |
| 1.96667 | -12.4089 |
| 2 | -13 |
| Quantity | Value | Unit |
|---|---|---|
| Endpoint potential difference | 13 | V |
| Horizontal-first path segment 1 work per charge | -8 | J/C |
| Horizontal-first path segment 2 work per charge | -5 | J/C |
| Vertical-first path segment 1 work per charge | -3 | J/C |
| Vertical-first path segment 2 work per charge | -10 | J/C |
| Total work per charge on either path | -13 | J/C |
| Closed-loop work per charge | 0 | J/C |
The segments differ, but the two complete electrostatic path integrals agree.
Integrate the field component along each displacement, not the field magnitude times path length. For the straight path (x,y) = (2t,t), E dot dr/dt = −(8A + 4B + 6)t, giving the same endpoint result. Time-dependent induction later requires revisiting the electrostatic premise.
Derive before checking. Retain contrasting captures and explain one assumption that could fail; calculated evidence is not actual laboratory work.
The segments differ, but the two complete electrostatic path integrals agree.
Integrate the field component along each displacement, not the field magnitude times path length. For the straight path (x,y) = (2t,t), E dot dr/dt = −(8A + 4B + 6)t, giving the same endpoint result. Time-dependent induction later requires revisiting the electrostatic premise.
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