Let E₁(r), E₂(r) and E₃(r) be the respective electric field at a distance r from a point charge Q, an infinitely long wire with constant linear charge density $λ$ , and an infinite plane with uniform surface charge density $σ$ .
If E₁(r₀) = E₂(r₀) = E₃(r₀) at a given distance r₀, then

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$r_{0} = \frac{4 λ}{πσ}$

$r_{0} = \frac{λ}{2 πσ}$

$r_{0} = \frac{λ}{4 πσ}$

$r_{0} = \frac{λ}{πσ}$

answer is A.

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Detailed Solution

$E_{1} = E_{2}$

$\Rightarrow \frac{1}{4 π ε_{0}} \frac{Q}{r_{0}^{2}} = \frac{1}{4 π ε_{0}} \frac{2 λ}{r_{0}} \Rightarrow Q = 2 λ r$

$E_{2} = E_{3}$

$\Rightarrow \frac{1}{4 π ε_{0}} \frac{2 λ}{r_{0}} = \frac{σ}{2 ε_{0}}$

$\Rightarrow r_{0} = \frac{λ}{σ π}$

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