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Let E1(r), E2(r) and E3(r) be the respec...

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

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)`

Text Solution

Verified by Experts

The correct Answer is:
C
`E_(1)=(Q)/(4piepsilon_(0)r^(2)),E_(2)=(lambda)/(2piepsilon_(0)r_(0)),E_(3)=(sigma)/(2epsilon_(0))`
`(Q)/(4piepsilon_(0)r_(0)^(2))=(lambda)/(2piepsilon_(0)r_(0))=(sigma)/(2epsilon_(0))`
`(Q)/(2r_(0))=lambdaimpliesr_(0)=Q//2lambda`
`(Q)/(4piepsilon_(0)r_(0)^(2))=(sigma)/(2epsilon_(0))impliesQ=2pisigmar_(0)^(2)`
`E_(1)((r_(0))/(2))=2E_(2)((r_(0))/(2))implies(Q)/(4piepsilon_(0)((r_(0))/(2))^(2))=(2lambda)/(2piepsilon_(0)((r_(0))/(2)))`
`implies(2lambdar_(0))/(4piepsilon_(0)(r_(0)^(2))/(4))=(2lambda)/(piepsilon_(0)r_(0))["as "r_(0)=(Q)/(2lambda)]`
L.H.S. = R.H.S.
`E_(2)((r_(0))/(2))=4E_(3)((r_(0))/(2))`
`implies (lambda)/(2piepsilon_(0).(r_(0))/(2))=(4sigma)/(2epsilon_(0))["as "(lambda)/(pir_(0))=sigma]`
L.H.S. `ne` R.H.S.
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