Let `E_1(r)`, `E_2(r)` and `E_3(r)` be the respectively electric field at a distance r from a point charge `Q`, an infinitely long wire with constant linear charge density `lambda`, and an infinite plane with uniform surface charge density `sigma`. If `E_1(r_0)=E_2(r_0)=E_3(r_0)` at a given distance `r_0`, then

Options

A. (a) `Q=4sigmapir_0^2`
B. (b) `r_0=(lambda)/(2pisigma)`
C. (c) `E_1(r_0//2)=2E_2(r_0//2)`
D. (d) `E_2(r_0//2)=4E_3(r_0//2)`

Answer

C

Explanation

`E_1=E_2` (Given)
`(1)/(4piin_0)=(Q)/(r_0^2)=(1)/(4piin_0)(2lambda)/(r_0)`
`:.` `Q=2lamgar_0` …(i)
`E_2=E_3` (Given)
`(1)/(4pi in_0)(2lambda)/(r_0)=(sigma)/(2in_0)impliesr_0=(lambda)/(sigmapi)`
`:.` (b) is incorrect
`E_1=E_3` (Given)
`:.` `(1)/(4pi in_0)(Q)/(r_0^2)=(sigma)/(2in_0)impliesQ=2pisigmar_0^2`
`:.` (a) is incorrect
Now `E_1(r_0//2)=(1)/(4piin_0)(4Q)/(r_0^2)`
`=(1)/(4piin_0)xx(4xx2lambdar_0)/(r_0^2)=(1)/(4piin_0)(8lambda)/(r_0)`
and `2E_2(r_0//2)=2[(1)/(4piin_0)(4lambda)/(r_0)]=(1)/(4piin_0)(8lambda)/(r_0)`
`:.` (c) is correct
`E_2(r_0//2)=(1)/(4piin_0)(2lambda)/(r_0//2)=(1)/(4piin_0)(4lambda)/(r_0)=(lambda)/(piin_0r_0)`
`4E_3(r_0//2)=(4sigma)/(2in_0)=(2sigma)/(in_0)=(2)/(in_0)xx(lambda)/(pir_0)`
`:.` (d) is incorrect.