A charge `10^(-9)` C is located at origin in free space and another charge Q at ( 2, 0 , 0) . If the x -component of the electric field at (3 ,1 , 1) is zero , calculate the value of Q .

Explanation

Here , For charge `10^(-9) C` ,
`V_(1) = (3 -0) hati + (1- 0) hatj + (1-0) hatk`
`implies V_(1) = 3 hati + hatj + hatk`
For change Q ,
`V_(2) = (3 - 2) hati + (1 - 0) hatj + (1-0) hatk`
`= hati + hatj + hatk`
Now `|V_(1)| = sqrt(11)`
`implies V_(1) = (3 hati + hatj + hatk)/(sqrt(11)) ` and `|V_(2)|= sqrt3`
`implies V_(2) = (hati + hatj + hatk)/(sqrt3)`
x-component of electric field due to `10^(-9) C` charge `E_(1) = (k 10^(-9))/((sqrt(11))^(2)) ((3hati)/(sqrt(11)) hati)`
x-component of electric field due to Q charge
`E_(2) = (kQ)/((sqrt3)^(2)) ((1)/(sqrt3) hati)`
According to the question , `|E_(1)| = |E_(2)|`
`(k 10^(-9))/((sqrt(11))^(2)) ((3)/(sqrt(11))) = (kQ)/(sqrt3)^(2) ((1)/(sqrt3))`
`(3 xx 10^(-9))/(11 xx sqrt(11)) = (Q)/( 3 sqrt3)`
`implies Q = (9 sqrt3 xx 10^(-9))/(11 sqrt(11))`
`implies Q = + 0.42 xx 10^(-9) C`