Worked example: A ring of radius R = 0.1 m carries Q = +5 nC uniformly. Find E on the axis at x = 0.05 m and at x = 1 m. Step 1: Use E = kQx/(R² + x²)^{3/2}. Step 2: At x = 0.05: R² + x² = 0.01 + 0.0025 = 0.0125; (0.0125)^{3/2} = 0.001397. Numerator: kQx = (8.988 × 10⁹)(5 × 10⁻⁹)(0.05) = 2.247. E = 2.247/0.001397 ≈ 1608 N/C along the axis away from the ring. Step 3: At x = 1 m: R² + x² ≈ 1.01, (1.01)^{3/2} ≈ 1.015; numerator kQx = 44.94; E ≈ 44.3 N/C. Check with point-charge approximation: kQ/x² = 44.94/1 = 44.9 N/C — matches, confirming the ring looks point-like from far away.