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The linear charge density on the circumference of a circle of radius a varies as λ = λ₀ cos θ. The total charge on it is ______.

Asked in GSEB Board July 2016 · Ring, arc and disc

Answer: (2) zero

Step-by-step solution

Given: λ = λ₀ cos θ on a circle of radius a.

Idea: add up dq = λ dl round the circle, with dl = a dθ.

Q = ∫₀^2πλ₀ cos θ a dθ = aλ₀[sin θ]₀^2π

Q = aλ₀(sin 2π - sin 0) = 0

The half of the ring where cos θ > 0 carries positive charge, and the other half carries an equal negative charge.

So the total charge is zero.

Why the other options are wrong

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