A uniformly charged square insulating plate of side d carries charge Q. A point charge q is placed on its normal at distance d/2 from centre. Force magnitude is Qq/(N ε₀d²). Find N. [See Image.]
Physics · Mixed Chapter
JEE Main Practice

Question

A uniformly charged square insulating plate of side d carries charge Q. A point charge q is placed on its normal at distance d/2 from centre. Force magnitude is Qq/(N ε₀d²). Find N. [See Image.]

G1 Physics · Accepted Answer

Step 1 — The axial field of a uniformly charged square is obtained by integrating the normal component of Coulomb field over its area. Step 2 — At normal distance z = d/2, consider a rectangle subtending solid angle Ω at charge; normal field E = σΩ/(4πε₀), with σ = Q/d². Step 3 — Solid angle for square at z=d/2 is Ω = 4 arctan[(d/2)²/(z√(z²+2(d/2)²))] = 4 arctan(1/√3) = 2π/3. Step 4 — E = (Q/d²)(2π/3)/(4πε₀) = Q/(6ε₀d²). Therefore force F = qE = Qq/(6ε₀d²); N = 6.