Separate a vector field from a signed test force

Define the passive probe assumption

The electric field is the force per positive test charge in the limit that the probe does not substantially disturb the sources. Writing E = F/q does not mean E depends on whichever test charge is chosen. At q = 0 use the independently specified source field rather than dividing zero force by zero charge.

Sum the source displacement vectors

A source q at rsource contributes E = kq(robs−rsource)/|robs−rsource|³. The denominator contains the cube because the numerator still contains a vector of length r. Adding these components implements superposition. Negative q reverses the contribution. Never sum magnitudes first when source directions differ.

Read maps and field lines honestly

A vector map attaches a local direction and scaled magnitude to sample points. A field-line diagram draws curves tangent to that field and uses line density only as a relative-strength convention. Lines are not trajectories of a massive particle, whose initial velocity also matters. The two graphs here are Ex(x,1) and Ey(x,1), not field lines. Singular source locations are excluded by sampling a line one metre above them.

Distinguish source materials

In electrostatic equilibrium the electric field inside conducting material is zero, and the surface field has no tangential component; otherwise mobile charge would keep moving. Excess charge resides at its surface. Insulators can retain assigned volume charge and nonzero interior fields. A hollow cavity, a conducting material region and an insulating volume are different domains; later spherical examples compare them explicitly.

Worked original case

At (0,1) m, sources +2 nC at (−1,0) and −2 nC at (1,0) have equal positive x contributions and cancelling y contributions. Ex≈12.7103 N/C, Ey = 0. A +1 nC probe feels +1.27103×10⁻⁸ N along x; reversing only the probe reverses its force.

Investigation

Compare right-source charges −2, 0 and +2 nC at a fixed probe. Construct equal-scale vector arrows at (0,1) before checking the component slices. Then vary only the probe through negative, zero and positive values; retain the unchanged field and changed force. Explain why the displayed component curves are not physical particle trajectories.

Public numerical transfer practice

Sources and evidence boundary

Original cases and teaching checked 30 October 2026 against the current official calculus-based electricity and magnetism framework and OpenStax electric fields. All displayed records are generated or supplied, not learner measurements. No charging, mains connection, high-voltage equipment, physical circuit or material handling is requested. Public feedback and retained self-reports are practice evidence, not independent mastery, performed laboratory time or AP authorization.