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Two charges q(1) and q(2) are placed at ...

Two charges `q_(1)` and `q_(2)` are placed at (0, 0, d) and (0, 0, - d) respectively. Find locus or points where the potential is zero.

Text Solution

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Let us talce a point of the required plane as (x, y, z). The two charges lies on z-axis at a separation of 2d.
The potential at the point P due to two charges is given by `V= V_(1)+V_(2)`
`:. 0 = (kq_(1))/(sqrt(x^(2)+y^(2)+(z-d)^(2)))+(kq^(2))/(sqrt(x^(2)+y^(2)+(z+d)^(2)))`
`:. (q_(1))/(sqrt(x^(2)+y^(2)+(z-d)^(2)))=-(q_(2))/(sqrt(x^(2)+y^(2)(z+d)^(2)))`
`(q_(1))/(q_(2))=- sqrt((x^(2)+y^(2)+(z-d)^(2))/(x^(2)+y^(2)+(z+d)^(2)))`
Compendo and dividendo
`sqrt(x^(2)+y^(2)+(z-d)^(2))`
`(q_(1)+q_(2))/(q_(1)-q_(2))=-(+sqrt(x^(2)+y^(2)+(z-d)^(2)))/(sqrt(x^(2)+y^(2)+(z-d)))`
Squaring both side
` (x^(2)+y^(2)+z^(2)-2zd+d^(2))`
`((q_(1)+q_(2))^(2))/((q_(1)-q_(2))^(2))=-(+(x^(2)+y^(2)+z^(2)+2zd+d^(2)))/((x^(2)+y^(2)+z^(2)-2zd+d^(2)))`
`-(x^(2)+y^(2)+z^(2)+2zd+d^(2))`
`=(2(x^(2)+y^(2)+z^(2)+d^(2)))/(2(2zd))`
`((q_(1)+q_(2))^(2))/((q_(1)-q_(2))^(2))=(x^(2)+y^(2)+z^(2)+d^(2))/(2zd)`
`:. x^(2)+y^(2)+z^(2)+2zd((q_(1)+q_(2))^(2))/((q_(1)-q_(2))^(2))+d^(2)=0`
This is the equation of sphere with centre ,
`(0,0,-2d[(q_(1)^(2)+q_(2)^(2))/(q_(1)^(2)-q_(2)^(2))])`
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