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

Answer

C

Explanation

We can write electric field intensities as follows :
`E_1(r_0)=1/(4piepsilon_0)Q/r_0^2`….(i)
`E_2(r_0)=lambda/(2piepsilon_0r_0)`….(ii)
`E_3(r_0)=sigma/(2epsilon_0)`…(iii)
`E_1(r_0)=E_2(r_0) rArr 1/(4piepsilon_0) Q/r_0^2 =lambda/(2piepsilon_0r_0) rArr Q/(2r_0) =lambda` ...(iv)
`E_2(r_0) =E_3(r_0) rArr lambda/(2piepsilon_0r_0)=sigma/(2epsilon_0) rArr lambda=pisigmar_0` ....(v)
`E_1(r_0)=E_3(r_0)rArr 1/(4piepsilon_0) Q/r_0^2 =sigma/(2epsilon_0) rArr Q=2pir_0^2 sigma`.....(vi)
From equation (vi) we can see that option (a) cannot be correct.
From equation (v) we can see that option (b) cannot be correct.
From equation (i) we can write
`E_1(r_0//2)=1/(4piepsilon_0)(4Q)/r_0^2 =Q/(piepsilon_0r_0^2)`...(vii)
From equation (ii) we can write
`E_2(r_0//2)=lambda/(piepsilon_0r_0)`...(viii)
Substituting the value of `lambda` from equation (iv) in equation (viii) we can write :
`2E_2(r_2//2)=(2 Q/(2r_0))/(piepsilon_0r_0)=Q/(piepsilon_0r_0^2)`....(ix)
From equation (vii) and (ix) we can see that `E_1(r_0//2)=2E_2(r_0//2)` , hence option (c ) is correct .
From equation (iii) we can write the following :
`4xxE_3(r_0//2)=(2sigma)/epsilon_0` ...(x)
From equation (v) we can substitute `sigma` in equation (x). We get the following :
`4xxE_3(r_0//2)=(2 lambda/(pir_0))/epsilon_0 =(2lambda)/(piepsilon_0r_0)`....(xi)
From equation (viii) and (xi) we can see that option (d) is not correct.