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Consider three charges q(1),q(2) and q(3...

Consider three charges `q_(1),q_(2)` and `q_(3)` each equal to `q`, at the vertices of an equilateral triangle of side l. What is the force on a charge Q placed at the centroid of the triangle?

Text Solution

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As shown in Fig , draw `AD _|_ BC`.
`:. AD = AB cos 30^(@) = (lsqrt(3))/(2)`

Distance AO of the centroid O from A
` = (2)/(3)AD = (2)/(3) (l sqrt(3))/(2) = (l)/(sqrt(3))`
`:.` Force on Q at O due to charge `q_(1) = q at A `
`F_(1) = (1)/(4pi in(0)) (Qq)/(11 sqrt(3)^(2)) = (3Qq)/(4pi in_(0) l^(2))`,along AO
Similary, force on Q due to charge `q_(2) = q at B`
`F_(1) = (3Qq)/(4pi in_(@)l^(2))` along BO
and force on Q due to charge `q_(3) = q at C`
`F_(3) = (3Qq)/(4pi in_(@) l^(2))` , along CO
Angle between forces `F_(2) and F_(3) = 120^(0)`
By parallelogram law, resultant of `F_(2)` and
`F_(3) = (3Qq)/(4pi in_(0) l^(2)) = 0`
`:.`Total force on `Q = (3Qq)/(4pi in(0) l^(2)) - (3Qq)/(4pi in(0) l^(2)) = 0`
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