Choose reference levels before taking an infinite-source limit
A potential reference valid for a finite localized source need not survive an infinite-source limit. For a uniform finite rod of half-length L and fixed λ=1 nC/m, the perpendicular-bisector potential with zero at infinity is the integral Kλ ds/√(s²+y²). It evaluates to 2Kλ asinh(L/y)=18 asinh(L/y) V.
Learning outcome: Compare L=1,3,5 m for the potentials at y=1 and2 m and their difference. Retain the increasing absolute potential separately from the converging difference. Explain why declaring V at infinity zero for the infinite line would contradict the integral.
Starting recap
Review oriented normals, definite integrals, source volume measures and partial derivatives as one-coordinate rates. Potential differences use an explicit reference and field work uses a signed line integral.
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For any finite L, the potential difference at y=1 m relative to itself is exactly zero. The infinite-line difference from y0=1 m to y=2 m is −18ln2≈−12.476649 V. For finite L the corresponding expression is 18[asinh(L/2)−asinh(L)], which approaches that result as L grows at fixed λ.
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Which reference statement is valid for an ideal infinite charged line?
Experiment
Compare L=1,3,5 m for the potentials at y=1 and2 m and their difference. Retain the increasing absolute potential separately from the converging difference. Explain why declaring V at infinity zero for the infinite line would contradict the integral.
Separate finite-source reference from infinite-line differences
Worked teaching model · synthetic values · Finite uniform rod with fixed lambda=1 nC/m, half-length L. V(y)=18 asinh(L/y) V for y>0 and reference zero at spatial infinity for each finite L. Compare V(y)−V(1). Infinite-line total potential with infinity reference is not finite; only a difference relative to finite y0 is used.
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Reference-subtracted finite and infinite-line comparisons
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: finite-rod potential difference (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Reference-subtracted finite and infinite-line comparisons
finite-rod potential difference
| perpendicular distance (m) | potential difference from distance 1 (V) |
|---|---|
| 0.5 | 12.12 |
| 0.53125 | 11.04452 |
| 0.5625 | 10.03235 |
| 0.59375 | 9.076714 |
| 0.625 | 8.171908 |
| 0.65625 | 7.313044 |
| 0.6875 | 6.495921 |
| 0.71875 | 5.716896 |
| 0.75 | 4.97279 |
| 0.78125 | 4.26081 |
| 0.8125 | 3.578495 |
| 0.84375 | 2.923656 |
| 0.875 | 2.294346 |
| 0.90625 | 1.68882 |
| 0.9375 | 1.105508 |
| 0.96875 | 0.5429956 |
| 1 | 0 |
| 1.03125 | -0.5246433 |
| 1.0625 | -1.031996 |
| 1.09375 | -1.523029 |
| 1.125 | -1.99863 |
| 1.15625 | -2.459619 |
| 1.1875 | -2.906748 |
| 1.21875 | -3.340711 |
| 1.25 | -3.762154 |
| 1.28125 | -4.171672 |
| 1.3125 | -4.569821 |
| 1.34375 | -4.957116 |
| 1.375 | -5.334039 |
| 1.40625 | -5.701041 |
| 1.4375 | -6.058542 |
| 1.46875 | -6.406938 |
| 1.5 | -6.746598 |
| 1.53125 | -7.077871 |
| 1.5625 | -7.401085 |
| 1.59375 | -7.716549 |
| 1.625 | -8.024555 |
| 1.65625 | -8.325379 |
| 1.6875 | -8.619281 |
| 1.71875 | -8.906509 |
| 1.75 | -9.187298 |
| 1.78125 | -9.461868 |
| 1.8125 | -9.730432 |
| 1.84375 | -9.993191 |
| 1.875 | -10.25034 |
| 1.90625 | -10.50205 |
| 1.9375 | -10.74851 |
| 1.96875 | -10.98987 |
| 2 | -11.2263 |
| 2.03125 | -11.45795 |
| 2.0625 | -11.68496 |
| 2.09375 | -11.90747 |
| 2.125 | -12.12562 |
| 2.15625 | -12.33952 |
| 2.1875 | -12.54932 |
| 2.21875 | -12.75511 |
| 2.25 | -12.95702 |
| 2.28125 | -13.15514 |
| 2.3125 | -13.3496 |
| 2.34375 | -13.54048 |
| 2.375 | -13.72788 |
| 2.40625 | -13.9119 |
| 2.4375 | -14.09262 |
| 2.46875 | -14.27013 |
| 2.5 | -14.44452 |
| 2.53125 | -14.61586 |
| 2.5625 | -14.78423 |
| 2.59375 | -14.94971 |
| 2.625 | -15.11236 |
| 2.65625 | -15.27226 |
| 2.6875 | -15.42948 |
| 2.71875 | -15.58408 |
| 2.75 | -15.73611 |
| 2.78125 | -15.88566 |
| 2.8125 | -16.03276 |
| 2.84375 | -16.17749 |
| 2.875 | -16.31989 |
| 2.90625 | -16.46003 |
| 2.9375 | -16.59794 |
| 2.96875 | -16.73369 |
| 3 | -16.86731 |
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 |
|---|---|---|
| Finite rod total charge | 6 | nC |
| Finite rod potential at distance 1 | 32.73204 | V |
| Finite rod potential at selected distance | 21.50574 | V |
| Finite rod potential difference | -11.2263 | V |
| Infinite-line finite-reference difference | -12.47665 | V |
| Finite minus infinite difference | 1.250351 | V |
An infinite line has a finite potential difference between finite radii, not a finite potential with zero at infinity.
Integrating Kλ ds/√(s²+y²) along a finite rod gives 2Kλ asinh(L/y). At fixed λ it diverges logarithmically as L grows. Subtracting the potential at y0=1 before taking the limit gives −2Kλ ln(y/y0), a finite difference. A logarithm must have a dimensionless argument. Gauge choice removes an additive constant, not the physical radial field.
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An infinite line has a finite potential difference between finite radii, not a finite potential with zero at infinity.
Integrating Kλ ds/√(s²+y²) along a finite rod gives 2Kλ asinh(L/y). At fixed λ it diverges logarithmically as L grows. Subtracting the potential at y0=1 before taking the limit gives −2Kλ ln(y/y0), a finite difference. A logarithm must have a dimensionless argument. Gauge choice removes an additive constant, not the physical radial field.
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