Practice portal › Electric Charge and Properties › Field of Continuous Charge Distributions
Asked in GUJCET 2008 · Ring, arc and disc
Given: λ = λ₀ cos²θ on a circle of radius a.
Idea: total charge Q = ∮λ dl, with dl = a dθ.
Q = ∫₀^2πλ₀ cos²θ a dθ = aλ₀∫₀^2π(1+cos 2θ)/2 dθ
Q = aλ₀[θ/2 + (sin 2θ)/4]₀^2π = π aλ₀
Check: the average of cos²θ over a full turn is 1/2, and 1/2×2π aλ₀ = π aλ₀.
So the total charge is π aλ₀.
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