Build a resultant field from source components

Establish the model and evidence

A field describes how a small probe would interact with fixed sources at each location. Its definition is E=F/q for a sufficiently small test charge; a positive probe feels force along E and a negative probe opposite E. The field belongs to the source configuration, not to the chosen sign of the probe. Making the probe large enough to rearrange the sources breaks that test-charge approximation.

At the observation point, draw one field arrow for each source. A positive source points away from itself, a negative source toward itself. Resolve each arrow into x and y components, add signed x components and signed y components separately, then find magnitude √(Ex²+Ey²). Adding individual magnitudes first discards cancellation and usually gives the wrong answer.

Two equal positive charges at symmetric locations provide an especially clear test. Above their midpoint, the horizontal components cancel while vertical components add. Reversing only the right charge changes which components cancel: vertical components cancel and horizontal components point from the positive source toward the negative source. Symmetry is evidence for a component cancellation, not a reason to set the entire field to zero.

The arrows in this activity all start at an abstract common origin so contributions can be compared with one scale. They are not field lines or a spatial map. A field line is tangent to the field direction at every location; it is not generally a charged particle’s trajectory, since the particle can have initial velocity. The profile below the arrows samples the x component along a horizontal line at the selected height.

Worked evidence

For two +2 nC sources at x=±0.20 m and a point (0,0.30) m, each source is √0.13 m away. Each contributes a vertical component k(2×10⁻⁹)(0.30)/(0.13)^(3/2)≈115.08 N/C. Net Ex=0 and net Ey≈230.16 N/C.

Predict and investigate

If only the right source changes from +2 to −2 nC, which net component will vanish?

Retain equal-positive and equal-opposite source cases at height 0.30 m. Reconstruct both resultants from the table. Then change the height to 0.50 m without changing charges and compare actual magnitudes rather than rescaled screen lengths.

Transfer to changed conditions

At another point Ex=−120 N/C and Ey=160 N/C. Find field magnitude and force components on a −2 nC probe.

Compare the worked transfer

The magnitude is 200 N/C. Multiplying each component by −2×10⁻⁹ C gives Fx=+2.4×10⁻⁷ N and Fy=−3.2×10⁻⁷ N. Its force magnitude is 4.0×10⁻⁷ N, opposite the field direction.

Explain the boundary

Explain why a negative test charge reverses its force but does not reverse the source field.

Construct a spatial field map from separate observations

Contribution arrows at one origin are not yet a spatial map. To make one, attach each resultant arrow to the location where the field is evaluated, use one length scale, and join smooth field lines only so that their local tangents follow those arrows. In a consistently drawn field-line diagram, closer line spacing represents stronger field; lines are not material threads and their arbitrary count is not a new physical unit.

Predict, sketch and check: use a single positive point source at the origin and stipulate field magnitude 100 N/C at radius r. At (r,0), (0,r) and (−r,0), draw equal arrows pointing respectively right, up and left. At (2r,0), draw a rightward arrow one quarter as long, representing 25 N/C. Reverse the source sign and reverse every arrow without changing lengths. A point source is justified only when its physical size is negligible for these observation distances; it is not a claim that an extended object has no geometry. (Current clauses 10.1.A.1.iv and 10.3.A.3.)

Sources and authorship

Physical relationship reference. Current AP Physics 2 framework, checked 30 October 2026. These cases, diagrams and numerical records are original. All experiments are bounded browser calculations or synthetic data, not measured laboratory evidence. Do not construct or handle charged capacitors, high-voltage sources or particle-beam equipment for these activities.

Clause-level source/action reconciliation checked 31 October 2026 against the linked current framework and Fall 2026 clarification. Added packets are original virtual comparisons; no new practical or learner-completion claim.