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Four points charges -q,+q,+q and -q are ...

Four points charges -q,+q,+q and -q are placed on y-axis at y=-2d, y=-d,y=+d and y=+2d, respectively. The magnitude of the electric field E at a point on the x-axis at x=D, with `Dgtgtd`, will behave as :

A

`E prop (1)/(D^(4))`

B

`E prop (1)/(D)`

C

`E prop (1)/(D^(2))`

D

`E prop (1)/(D^(3))`

Text Solution

Verified by Experts

The correct Answer is:
A
Electric field at `p = 2E_(1) cos theta_(1) -2 E_(1) cos theta_(2)`

`= (3kq)/((d^(2) + D^(2))) xx (D)/((d^(2) + D^(2))^(1//2)) - (2kq)/([(2d)^(2) + D^(2)]) xx (D)/([(2d)^(2) + D^(2)]^(1//2))`
`= 2kqD [(d^(2) + D^(2))^(-3//2) - (4 d^(2) + d^(2))^(-3//2)]`
`= (2KqD)/(D^(3)) [(a + (d^(2))/(D^(2)))^(-3//3) - (1 + (4d^(2))/(D^(2)))^(-3//2)]`
Applying binominal approximation `:. d lt lt D`
`= (2KqD)/(D^(3)) [1 - (3)/(2) (d^(2))/(D^(2)) - (1 - (3 xx 4 d^(2))/(2D^(2)))]`
`= (2KqD)/(D^(3)) [(12)/(2) (d^(2))/(D^(2)) - (3)/(2) (d^(2))/(D^(2))]`
`= (9Kqd^(2))/(D^(4))`
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