Differentiate a projected ring field
Every element of a uniform ring has the same distance to an axial observer, but only its axial projection survives the vector sum. With radius R and axial coordinate z, that projection contributes an additional z/√(R²+z²) factor. The axial field is KQz/(R²+z²)^(3/2), not simply KQ/(R²+z²).
Learning outcome: At a fixed R, compare center, z/R=0.5,1,2,3 and compute the analytic maximum between supported grid points. Retain derivative signs and distinguish a grid maximum from the true stationary point. Double R and test both the location and magnitude scaling.
Starting recap
Review signed vector components, definite integrals, radians, density units and elementary derivatives. Distinguish observer coordinates from source integration coordinates.
All lessons are open directly. Your saved experiments remain when you leave and return.
Try
For Q=2 nC and R=1 m, the field at z=1 m is 9/√2≈6.363961 N/C. The maximum occurs instead at z=1/√2 m and equals 12/√3≈6.928203 N/C. The derivative at z=1 m is negative, consistent with lying beyond the maximum.
Open the experimentPredict
For a positive uniformly charged ring, the positive axial maximum occurs at:
Experiment
At a fixed R, compare center, z/R=0.5,1,2,3 and compute the analytic maximum between supported grid points. Retain derivative signs and distinguish a grid maximum from the true stationary point. Double R and test both the location and magnitude scaling.
Derive the stationary axial field of a charged ring
Worked teaching model · synthetic values · Uniform positive 2 nC ring in the xy plane; observe z=bR≥0. K=9×10^9. A zero field at z=0 is a vector cancellation, not zero charge. Point-charge comparison is withheld at z=0 rather than assigning a finite value to its singular expression.
Reset keeps your prediction, captures and practice history.
Ring axial field and its stationary maximum
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: exact axial ring field (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Ring axial field and its stationary maximum
exact axial ring field
| axial coordinate (m) | axial field (N/C) |
|---|---|
| 0 | 0 |
| 0.05 | 0.8966355 |
| 0.1 | 1.773334 |
| 0.15 | 2.611372 |
| 0.2 | 3.394318 |
| 0.25 | 4.108839 |
| 0.3 | 4.745194 |
| 0.35 | 5.297379 |
| 0.4 | 5.762959 |
| 0.45 | 6.142673 |
| 0.5 | 6.439876 |
| 0.55 | 6.659914 |
| 0.6 | 6.809503 |
| 0.65 | 6.896162 |
| 0.7 | 6.927733 |
| 0.75 | 6.912 |
| 0.8 | 6.856409 |
| 0.85 | 6.767879 |
| 0.9 | 6.652688 |
| 0.95 | 6.516421 |
| 1 | 6.363961 |
| 1.05 | 6.199516 |
| 1.1 | 6.026661 |
| 1.15 | 5.848396 |
| 1.2 | 5.667206 |
| 1.25 | 5.485127 |
| 1.3 | 5.303804 |
| 1.35 | 5.124545 |
| 1.4 | 4.948379 |
| 1.45 | 4.776094 |
| 1.5 | 4.608279 |
| 1.55 | 4.445361 |
| 1.6 | 4.287632 |
| 1.65 | 4.135275 |
| 1.7 | 3.988385 |
| 1.75 | 3.846985 |
| 1.8 | 3.711045 |
| 1.85 | 3.58049 |
| 1.9 | 3.455212 |
| 1.95 | 3.335078 |
| 2 | 3.219938 |
| 2.05 | 3.109627 |
| 2.1 | 3.003972 |
| 2.15 | 2.902798 |
| 2.2 | 2.805924 |
| 2.25 | 2.713172 |
| 2.3 | 2.624366 |
| 2.35 | 2.539331 |
| 2.4 | 2.457897 |
| 2.45 | 2.379901 |
| 2.5 | 2.305184 |
| 2.55 | 2.233591 |
| 2.6 | 2.164977 |
| 2.65 | 2.0992 |
| 2.7 | 2.036125 |
| 2.75 | 1.975624 |
| 2.8 | 1.917574 |
| 2.85 | 1.861858 |
| 2.9 | 1.808365 |
| 2.95 | 1.756989 |
| 3 | 1.70763 |
| 3.05 | 1.660192 |
| 3.1 | 1.614586 |
| 3.15 | 1.570724 |
| 3.2 | 1.528527 |
| 3.25 | 1.487915 |
| 3.3 | 1.448817 |
| 3.35 | 1.411163 |
| 3.4 | 1.374887 |
| 3.45 | 1.339926 |
| 3.5 | 1.306221 |
| 3.55 | 1.273716 |
| 3.6 | 1.242358 |
| 3.65 | 1.212096 |
| 3.7 | 1.182882 |
| 3.75 | 1.154671 |
| 3.8 | 1.127419 |
| 3.85 | 1.101084 |
| 3.9 | 1.075629 |
| 3.95 | 1.051016 |
| 4 | 1.02721 |
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 |
|---|---|---|
| Axial location | 1 | m |
| Axial field | 6.363961 | N/C |
| Axial field derivative | -3.181981 | N/C/m |
| Positive axial maximum location | 0.7071068 | m |
| Positive axial maximum field | 6.928203 | N/C |
| Point-charge relative overestimate | 1.828427 | ratio |
Differentiating the projection factor as well as the inverse-square distance locates the maximum.
Each ring element has the same distance √(R²+z²), and radial components cancel. Ez=KQz/(R²+z²)^(3/2), so dEz/dz=KQ(R²−2z²)/(R²+z²)^(5/2). The positive maximum occurs at R/√2, not at the center or at R. Far away, Ez/(KQ/z²)=[1+(R/z)²]^(-3/2).
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Differentiating the projection factor as well as the inverse-square distance locates the maximum.
Each ring element has the same distance √(R²+z²), and radial components cancel. Ez=KQz/(R²+z²)^(3/2), so dEz/dz=KQ(R²−2z²)/(R²+z²)^(5/2). The positive maximum occurs at R/√2, not at the center or at R. Far away, Ez/(KQ/z²)=[1+(R/z)²]^(-3/2).
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
Understand
Apply
Public practice checks a specific claim. It does not establish independent mastery.
Which geometry, calibration or source assumption could fail while an exact arithmetic fit remains possible?
Continue
Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.
Skipping retains experiments and awards no mastery.
Save a science backup or import one
Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.
A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.
Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.
Complete file sets and course backups
A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.
The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.
0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.
Science storage: loading. No account or remote computation is required.