Derive local uniformity instead of matching one amplitude
An on-axis loop integral cancels transverse components while retaining the common axial projection. One loop of radius R and current I gives Bz=μ0IR²/[2(R²+z²)^(3/2)]. Two identical, same-current coaxial loops centered at ±d/2 add shifted copies of this result. In the workbench R=1 m and I=2 A.
Learning outcome: Compare separation ratios 0.5,1,1.5 at z/R=0,0.25,0.5. Retain center amplitude, second derivative and normalized departure as separate evidence. Explain why the zero-curvature profile still changes at finite distance.
Starting recap
Review cross products, source integration measures, line integrals, differential initial states and signed work. Distinguish symmetry premises from integral laws.
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At d=R=1 m, the center field is 2μ0/(1.25)^(3/2)≈1.798352 µT and the second derivative is exactly zero. At separation d=0.5 m the curvature is negative; at d=1.5 m it is positive. All three center first derivatives vanish, so first-derivative symmetry alone does not select the flat-profile spacing.
Open the experimentPredict
What distinguishes d=R from other symmetric loop separations?
Experiment
Compare separation ratios 0.5,1,1.5 at z/R=0,0.25,0.5. Retain center amplitude, second derivative and normalized departure as separate evidence. Explain why the zero-curvature profile still changes at finite distance.
Compare center strength with spatial uniformity
Worked teaching model · synthetic values · Two identical coaxial loops radius 1 m, current 2 A each with the same axial moment, centered at z=±d/2. On-axis Biot–Savart field only. d=R cancels the center second derivative; this local condition does not make an exactly uniform field everywhere.
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Paired-loop axial field normalized to its center
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: same-current coaxial loops (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Paired-loop axial field normalized to its center
same-current coaxial loops
| z divided by radius | field divided by center field |
|---|---|
| -1 | 0.6192642 |
| -0.975 | 0.6384682 |
| -0.95 | 0.6577861 |
| -0.925 | 0.6771575 |
| -0.9 | 0.6965178 |
| -0.875 | 0.7157982 |
| -0.85 | 0.734926 |
| -0.825 | 0.7538258 |
| -0.8 | 0.7724203 |
| -0.775 | 0.790631 |
| -0.75 | 0.8083796 |
| -0.725 | 0.825589 |
| -0.7 | 0.8421847 |
| -0.675 | 0.8580964 |
| -0.65 | 0.8732587 |
| -0.625 | 0.8876131 |
| -0.6 | 0.901109 |
| -0.575 | 0.9137044 |
| -0.55 | 0.9253677 |
| -0.525 | 0.9360776 |
| -0.5 | 0.9458242 |
| -0.475 | 0.954609 |
| -0.45 | 0.9624448 |
| -0.425 | 0.9693558 |
| -0.4 | 0.9753764 |
| -0.375 | 0.9805509 |
| -0.35 | 0.984932 |
| -0.325 | 0.9885798 |
| -0.3 | 0.9915603 |
| -0.275 | 0.9939433 |
| -0.25 | 0.9958016 |
| -0.225 | 0.9972085 |
| -0.2 | 0.9982365 |
| -0.175 | 0.9989554 |
| -0.15 | 0.9994311 |
| -0.125 | 0.9997235 |
| -0.1 | 0.9998861 |
| -0.075 | 0.9999638 |
| -0.05 | 0.9999928 |
| -0.025 | 0.9999996 |
| 0 | 1 |
| 0.025 | 0.9999996 |
| 0.05 | 0.9999928 |
| 0.075 | 0.9999638 |
| 0.1 | 0.9998861 |
| 0.125 | 0.9997235 |
| 0.15 | 0.9994311 |
| 0.175 | 0.9989554 |
| 0.2 | 0.9982365 |
| 0.225 | 0.9972085 |
| 0.25 | 0.9958016 |
| 0.275 | 0.9939433 |
| 0.3 | 0.9915603 |
| 0.325 | 0.9885798 |
| 0.35 | 0.984932 |
| 0.375 | 0.9805509 |
| 0.4 | 0.9753764 |
| 0.425 | 0.9693558 |
| 0.45 | 0.9624448 |
| 0.475 | 0.954609 |
| 0.5 | 0.9458242 |
| 0.525 | 0.9360776 |
| 0.55 | 0.9253677 |
| 0.575 | 0.9137044 |
| 0.6 | 0.901109 |
| 0.625 | 0.8876131 |
| 0.65 | 0.8732587 |
| 0.675 | 0.8580964 |
| 0.7 | 0.8421847 |
| 0.725 | 0.825589 |
| 0.75 | 0.8083796 |
| 0.775 | 0.790631 |
| 0.8 | 0.7724203 |
| 0.825 | 0.7538258 |
| 0.85 | 0.734926 |
| 0.875 | 0.7157982 |
| 0.9 | 0.6965178 |
| 0.925 | 0.6771575 |
| 0.95 | 0.6577861 |
| 0.975 | 0.6384682 |
| 1 | 0.6192642 |
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 |
|---|---|---|
| Center axial field | 1.79835e-6 | T |
| Observer axial field | 1.70093e-6 | T |
| Relative departure from center | -0.05417581 | ratio |
| Center second axial derivative | 0 | T/m² |
| Center first axial derivative by symmetry | 0 | T/m |
Symmetry removes the first derivative for every separation; only a special separation removes the second.
Integrating one loop cancels transverse components and gives Bz=mu0 I R²/[2(R²+z²)^(3/2)]. Superpose two shifted loops. At their midpoint Bprime=0 by reflection symmetry. Differentiation gives Bsecond=3 mu0 I R²(d²−R²)/(R²+d²/4)^(7/2), so d=R cancels the quadratic term. Higher-order terms remain: matching a center value or canceling one derivative does not establish uniformity over an arbitrary region.
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Symmetry removes the first derivative for every separation; only a special separation removes the second.
Integrating one loop cancels transverse components and gives Bz=mu0 I R²/[2(R²+z²)^(3/2)]. Superpose two shifted loops. At their midpoint Bprime=0 by reflection symmetry. Differentiation gives Bsecond=3 mu0 I R²(d²−R²)/(R²+d²/4)^(7/2), so d=R cancels the quadratic term. Higher-order terms remain: matching a center value or canceling one derivative does not establish uniformity over an arbitrary region.
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