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Let E1(r), E2(r) and E3(r) be the electric fields at distance r from (i) a point charge Q, (ii) an infinitely long line charge of linear density lambda, and (iii) an infinite plane sheet of surface charge density sigma, respectively. Given E1(r0) = E2(r0) = E3(r0) at a particular distance r0, which statement is correct?

1. Q = 4*pi*sigma*r0²
2. r0 = lambda / (2*pi*sigma)
3. E1(r0/2) = 2*E2(r0/2)
4. E2(r0/2) = 4*E3(r0/2)

Correct answer: r0 = lambda / (2*pi*sigma)

Solution

E2(r0) = lambda/(2*pi*epsilon0*r0) and E3 = sigma/(2*epsilon0) (independent of r). Setting them equal: lambda/(2*pi*epsilon0*r0) = sigma/(2*epsilon0), which gives r0 = lambda/(2*pi*sigma). Checking the distance scalings: E1 ~ 1/r², E2 ~ 1/r, E3 = const. At r0/2, E1 quadruples, E2 doubles, E3 unchanged, so E1(r0/2) = 4*E2... no; and E2(r0/2) = 2*E3..., not 4. Hence only r0 = lambda/(2*pi*sigma) is correct.

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