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four charges each equal to Q are placed at the four corners of a square and a charge q is placed at its centre of the square. If the system is in equilibrium then the value of q is
 

Asked by dilipdevesh1170 13th June 2016, 3:52 PM
Answered by Expert
Answer:
 
 
begin mathsize 12px style If space each space charge space experiences space no space force space then space the space system space is space in space equilibrium.
From space the space fihure space it space is space clear space that space the space charge space apostrophe straight q apostrophe space at space the space cnetre space would space be space in space equilibrium space for space any space value space of space straight q.
Let space AB equals BC equals CD equals DA equals straight a.
From space the space figure comma space we space get
DB equals square root of 2 straight a space and space OB equals fraction numerator straight a over denominator square root of 2 end fraction
straight F subscript BA equals straight F subscript BC equals fraction numerator straight Q squared over denominator 4 πε subscript straight o straight a squared end fraction
straight F subscript BD equals fraction numerator straight Q squared over denominator 4 πε subscript straight o open parentheses square root of 2 straight a close parentheses squared end fraction equals fraction numerator straight Q squared over denominator 4 πε subscript straight o left parenthesis 2 straight a right parenthesis squared end fraction
straight F subscript BO equals fraction numerator qQ over denominator 4 πε subscript straight o open parentheses begin display style fraction numerator straight a over denominator square root of 2 end fraction end style close parentheses squared end fraction equals negative fraction numerator 2 qQ over denominator 4 πε subscript straight o straight a squared end fraction
The space net space force space acting space on space minus straight Q space at space point space straight B space will space be space zero space if space
straight F subscript BA cos space 45 space plus space straight F subscript BC cos space 45 space plus space straight F subscript BD space plus space straight F subscript BO equals 0
rightwards double arrow fraction numerator straight Q squared over denominator 4 πε subscript straight o straight a squared end fraction cross times fraction numerator 1 over denominator square root of 2 end fraction plus fraction numerator straight Q squared over denominator 4 πε subscript straight o straight a squared end fraction cross times fraction numerator 1 over denominator square root of 2 end fraction plus fraction numerator straight Q squared over denominator 4 πε subscript straight o left parenthesis 2 straight a right parenthesis squared end fraction minus fraction numerator 2 qQ over denominator 4 πε subscript straight o straight a squared end fraction equals 0
On space solving space comma space we space get
rightwards double arrow straight q equals straight Q over 4 open parentheses 1 plus 2 square root of 2 close parentheses
end style
Answered by Expert 14th June 2016, 12:04 PM
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