Recover a vector field and verify path-independent work
Electric potential is a scalar whose spatial variation determines a vector field. A partial derivative changes one coordinate while holding the others fixed. For V=a(x²−y²)+bxy+7 V in the stated SI convention, Ex=−2ax−by and Ey=2ay−bx. The negative sign means a positive test charge accelerates toward decreasing potential if electric force is the only force.
Learning outcome: Derive both components before comparing the two paths for at least two coefficient pairs. Retain endpoint potentials, signed work and the effect of adding a constant reference offset. At one point choose a vector perpendicular to E and check that its first-order potential change vanishes.
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 a=2,b=1, the endpoint potential is 3 V versus 7 V at the origin. Thus ΔV=−4 V and electric work is 4 J/C by either path. At the endpoint E=(−6,7) N/C. A −2 nC test charge has potential-energy change +8 nJ and electric work −8 nJ for this displacement.
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For a=2,b=1, what is electric work per positive unit charge from origin to (1,2)?
Experiment
Derive both components before comparing the two paths for at least two coefficient pairs. Retain endpoint potentials, signed work and the effect of adding a constant reference offset. At one point choose a vector perpendicular to E and check that its first-order potential change vanishes.
Derive a vector field and check two work paths
Worked teaching model · synthetic values · Potential V=a(x²−y²)+bxy+7 V in a plane, with dimensioned coefficients. Field E=−gradient V. Compare work per positive unit charge from (0,0) to (1,2) along two axis-aligned paths. Gauge constant7 affects V but neither E nor potential differences.
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Two paths between the same potential endpoints
Spatial axes use the same pixels per physical unit, preserving geometry. 1: east then north (solid).2: north then east (dashed). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Two paths between the same potential endpoints
east then north
| east coordinate (m) | north coordinate (m) |
|---|---|
| 0 | 0 |
| 1 | 0 |
| 1 | 2 |
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.
north then east
| east coordinate (m) | north coordinate (m) |
|---|---|
| 0 | 0 |
| 0 | 2 |
| 1 | 2 |
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 |
|---|---|---|
| Potential at origin | 7 | V |
| Potential at endpoint | 3 | V |
| Endpoint east field | -6 | N/C |
| Endpoint north field | 7 | N/C |
| East-then-north work per charge | 4 | J/C |
| North-then-east work per charge | 4 | J/C |
| Potential change | -4 | V |
| Endpoint field magnitude | 9.219544 | N/C |
A scalar potential supplies every field component through a negative partial derivative, not through its value alone.
Ex=−2ax−by and Ey=2ay−bx. Integrating the first path gives −a+(4a−2b); the second gives 4a+(−a−2b), both 3a−2b=−Delta V. The additive 7 V cancels. A tangent displacement along an equipotential has gradient V dot dr=0 and therefore zero electric work per charge, even when the local field is nonzero.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
A scalar potential supplies every field component through a negative partial derivative, not through its value alone.
Ex=−2ax−by and Ey=2ay−bx. Integrating the first path gives −a+(4a−2b); the second gives 4a+(−a−2b), both 3a−2b=−Delta V. The additive 7 V cancels. A tangent displacement along an equipotential has gradient V dot dr=0 and therefore zero electric work per charge, even when the local field is nonzero.
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