Compare exact dipole geometry with a far-field expansion
Two equal and opposite charges define a dipole moment p=2qd pointing from negative to positive charge when their positions are −d and +d on the x axis. Holding p fixed while varying d requires changing q. Equal moment therefore does not imply equal individual charges or identical near fields.
Learning outcome: Keep p and r fixed while reducing d, retaining the required q and both exact fields. Compare signed and magnitude errors separately. Then explain why a distant field measurement of p alone cannot uniquely recover q and d.
Starting recap
Review signed vector components, definite integrals, radians, density units and elementary derivatives. Distinguish observer coordinates from source integration coordinates.
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For p=1 nC m, d=0.5 m and r=2 m, q=1 nC. The exact axial field is 2.56 N/C while its leading approximation is 2.25 N/C, a signed relative error −0.12109375. The exact equatorial field is −9/(4.25)^(3/2) N/C, not simply half the exact axial magnitude.
Open the experimentPredict
At fixed moment, the leading positive-axis far field scales with observer distance as:
Experiment
Keep p and r fixed while reducing d, retaining the required q and both exact fields. Compare signed and magnitude errors separately. Then explain why a distant field measurement of p alone cannot uniquely recover q and d.
Separate exact dipole geometry from its far-field approximation
Worked teaching model · synthetic values · Charges ±q at x=±d have fixed moment p=2qd=1 nC m pointing +x. Positive charge is at +d. Observer is either axial x=r>d or equatorial (0,r). K=9×10^9. As d changes q changes; this is a prescribed mathematical source family, not a construction or radiation model.
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Exact versus leading axial dipole scaling
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: exact fixed-moment source pair (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Exact versus leading axial dipole scaling
exact fixed-moment source pair
| axial distance (m) | east field (N/C) |
|---|---|
| 1 | 32 |
| 1.0375 | 27.34472 |
| 1.075 | 23.59305 |
| 1.1125 | 20.52867 |
| 1.15 | 17.99599 |
| 1.1875 | 15.88083 |
| 1.225 | 14.0979 |
| 1.2625 | 12.58247 |
| 1.3 | 11.28472 |
| 1.3375 | 10.16581 |
| 1.375 | 9.195102 |
| 1.4125 | 8.34819 |
| 1.45 | 7.605435 |
| 1.4875 | 6.950899 |
| 1.525 | 6.371542 |
| 1.5625 | 5.856616 |
| 1.6 | 5.3972 |
| 1.6375 | 4.985845 |
| 1.675 | 4.61629 |
| 1.7125 | 4.283246 |
| 1.75 | 3.982222 |
| 1.7875 | 3.709385 |
| 1.825 | 3.461447 |
| 1.8625 | 3.23558 |
| 1.9 | 3.029337 |
| 1.9375 | 2.840595 |
| 1.975 | 2.667504 |
| 2.0125 | 2.50845 |
| 2.05 | 2.362015 |
| 2.0875 | 2.226952 |
| 2.125 | 2.102162 |
| 2.1625 | 1.986672 |
| 2.2 | 1.879619 |
| 2.2375 | 1.780236 |
| 2.275 | 1.68784 |
| 2.3125 | 1.601818 |
| 2.35 | 1.521623 |
| 2.3875 | 1.446765 |
| 2.425 | 1.376801 |
| 2.4625 | 1.311332 |
| 2.5 | 1.25 |
| 2.5375 | 1.192479 |
| 2.575 | 1.138474 |
| 2.6125 | 1.087719 |
| 2.65 | 1.039969 |
| 2.6875 | 0.9950032 |
| 2.725 | 0.9526197 |
| 2.7625 | 0.9126339 |
| 2.8 | 0.874877 |
| 2.8375 | 0.8391944 |
| 2.875 | 0.8054444 |
| 2.9125 | 0.7734967 |
| 2.95 | 0.7432316 |
| 2.9875 | 0.7145388 |
| 3.025 | 0.6873167 |
| 3.0625 | 0.6614715 |
| 3.1 | 0.6369165 |
| 3.1375 | 0.6135717 |
| 3.175 | 0.5913628 |
| 3.2125 | 0.570221 |
| 3.25 | 0.5500826 |
| 3.2875 | 0.5308884 |
| 3.325 | 0.5125832 |
| 3.3625 | 0.4951158 |
| 3.4 | 0.4784386 |
| 3.4375 | 0.462507 |
| 3.475 | 0.4472796 |
| 3.5125 | 0.4327177 |
| 3.55 | 0.418785 |
| 3.5875 | 0.4054478 |
| 3.625 | 0.3926744 |
| 3.6625 | 0.3804351 |
| 3.7 | 0.3687022 |
| 3.7375 | 0.3574495 |
| 3.775 | 0.3466527 |
| 3.8125 | 0.3362888 |
| 3.85 | 0.3263361 |
| 3.8875 | 0.3167744 |
| 3.925 | 0.3075845 |
| 3.9625 | 0.2987485 |
| 4 | 0.2902494 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Positive source charge | 1 | nC |
| Exact axial east field | 2.56 | N/C |
| Leading axial dipole approximation | 2.25 | N/C |
| Exact equatorial east field | -1.02721 | N/C |
| Leading equatorial dipole approximation | -1.125 | N/C |
| Axial approximation relative signed error | -0.1210938 | ratio |
| Equatorial approximation relative signed error | 0.09519993 | ratio |
A shared dipole moment fixes the leading distant field, not the exact near field.
For axial r>d, Kq[(r−d)^(-2)−(r+d)^(-2)]=2Kpr/(r²−d²)². On the equatorial axis the transverse components cancel and Ex=−Kp/(r²+d²)^(3/2). Their leading far fields are 2Kp/r³ and −Kp/r³. Use the exact expressions to quantify when discarding powers of d/r is acceptable.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
A shared dipole moment fixes the leading distant field, not the exact near field.
For axial r>d, Kq[(r−d)^(-2)−(r+d)^(-2)]=2Kpr/(r²−d²)². On the equatorial axis the transverse components cancel and Ex=−Kp/(r²+d²)^(3/2). Their leading far fields are 2Kp/r³ and −Kp/r³. Use the exact expressions to quantify when discarding powers of d/r is acceptable.
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