Integrate ring projections before taking limits

Choose the axis before simplifying

Place a uniform thin ring in the xy plane and observe at (0,0,z). Every source element is at distance sqrt(R²+z²). Equal opposite azimuthal elements cancel their transverse field components, but their axial projections add. This is a geometrical symmetry argument, not permission to replace all extended sources by points.

Carry the projection through the integral

The axial element is dEz = k z dq/(R²+z²)^(3/2). The factors involvingR and z are constant over the ring, so integratingdq yieldsQ. Thus Ez = kQz/(R²+z²)^(3/2). The sign ofz determines the direction for positiveQ. The result applies only on the axis; it does not provide an off-axis field map.

Compare center and distant behavior

At z = 0 the result vanishes. When |z| is much larger thanR, the leading magnitude approaches kQ/z² with the appropriate axial sign. Near the center, the field is approximately(kQ/R³)z. Neither limit is the exact function at every distance; retain the ratio |z|/R when arguing an approximation.

Optional derivative challenge

For positivez, differentiating gives dEz/dz = kQ(R²−2z²)/(R²+z²)^(5/2). The derivative vanishes at R/sqrt 2 and changes from positive to negative, establishing the maximum rather than merely finding a stationary point. This calculus inference supplements the required field integration. A zero derivative at the maximum does not mean the field itself is zero.

Worked original case

WithQ = 2 nC,R = 1 m,z = 1 m, Ez = kQ/(2sqrt 2)≈6.35516 N/C. At z = 0 every transverse contribution cancels and the axial projection is zero, despite nonzero charge. The positive maximum lies at z = R/sqrt 2 and is about 6.91862 N/C.

Investigation

At fixed total Q, compare radii.5,1,2 m and equal axial coordinates. Predict the sign across the ring plane, the center value and far-field scale before plotting. Independently differentiate the analytic field if choosing the optional maximum-location challenge; distinguish that extension from the core projection integral.

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.