Electric Field at Centre of Ring with Non-Uniform Charge Distribution. Four quadrants carry linear charge densities: +2λ, −2λ, +λ, −λ. Find electric field at centre.

For a quarter-circle arc of linear charge density λ and radius R, the field at the centre has magnitude:

E_quadrant = (√2 λ) / (4πε₀R)

Direction: along the bisector of the quadrant, pointing away from arc (for +λ) or toward arc (for −λ).

Resolving all four quadrant fields into x and y components:

Ex = [−2 − 2 + 1 + 1] × (λK/√2) = −2λK/√2

Ey = [−2 + 2 + 1 − 1] × (λK/√2) = 0

Magnitude:

|E| = 2λK/√2 = λ / (2πε₀R)

Answer: E = λ / (2πε₀R), directed along the negative x-axis