3 identical charged spheres each having charge 'q' and mass m are suspended from same point through a 3 equal strings of length 'l' . if they are in equilibrium at the vertices of an equilateral triangle of side length 'a'. find 'a'.

Asked by Sunil Soni | 16th Jan, 2016, 08:56: AM

Expert Answer:

Consider three indentical charged spheres each having charge q and mass m are suspended through a three equal strings of length l.
Let they are in equilibrium at the vertices of an equilateral triangle.
Total height of the triangle 'h' is begin mathsize 12px style fraction numerator square root of 3 straight a over denominator 2 end fraction end style and centroid of an equilateral triangle is begin mathsize 12px style fraction numerator straight a over denominator square root of 3 end fraction end style
 
Consider the right angle traingle AEC
begin mathsize 12px style cos space straight theta space equals space fraction numerator straight a over denominator square root of 3 straight l end fraction end style
From the pythagoras theorem side AE is
begin mathsize 12px style AC to the power of 2 space end exponent equals space AE squared space plus space EC squared straight l squared equals AE squared space plus space open parentheses fraction numerator straight a over denominator square root of 3 end fraction close parentheses squared AE squared equals straight l squared minus open parentheses fraction numerator straight a over denominator square root of 3 end fraction close parentheses squared AE equals square root of straight l squared minus open parentheses fraction numerator straight a over denominator square root of 3 end fraction close parentheses squared end root end style
From the figure,
 
begin mathsize 12px style tan space straight theta space equals space AE over EC equals fraction numerator square root of straight l squared minus open parentheses fraction numerator straight a over denominator square root of 3 end fraction close parentheses squared end root over denominator open parentheses fraction numerator straight a over denominator square root of 3 end fraction close parentheses end fraction end style
begin mathsize 12px style mg space equals space Tcos space straight alpha mg space equals space space Tcos space left parenthesis 90 minus straight theta right parenthesis mg equals Tsin space straight theta space space space space space space left parenthesis straight i right parenthesis end style
Since, the three charges are identical possesing repulsive force outwards as shown in diagram
from the equillibrium system.
begin mathsize 12px style 2 straight F space cos space 30 degree equals space straight T space sinα straight F equals Tsin left parenthesis 90 minus straight theta right parenthesis rightwards double arrow straight F equals Tcos space straight theta space space space space space space space space space space space space left parenthesis ii right parenthesis space end style
From (i) and (ii)
begin mathsize 12px style tan space straight theta space equals space straight F over mg equals fraction numerator straight q squared over denominator 4 πε subscript 0 straight a squared mg end fraction tan space straight theta space equals fraction numerator square root of straight l squared minus open parentheses fraction numerator straight a over denominator square root of 3 end fraction close parentheses squared end root over denominator open parentheses fraction numerator straight a over denominator square root of 3 end fraction close parentheses end fraction equals fraction numerator straight q squared over denominator 4 πε subscript 0 straight a squared mg end fraction rightwards double arrow fraction numerator square root of straight l squared minus open parentheses fraction numerator straight a over denominator square root of 3 end fraction close parentheses squared end root over denominator open parentheses fraction numerator straight a over denominator square root of 3 end fraction close parentheses end fraction equals fraction numerator straight q squared over denominator 4 πε subscript 0 straight a squared mg end fraction  rightwards double arrow fraction numerator straight l squared square root of 3 minus open parentheses straight a close parentheses squared over denominator straight a squared end fraction equals fraction numerator straight q to the power of 4 over denominator 4 squared straight pi squared straight epsilon subscript 0 superscript 1 straight a squared mg end fraction  rightwards double arrow straight a space equals square root of straight l squared square root of 3 minus fraction numerator straight q to the power of 4 over denominator 8 straight pi squared straight epsilon subscript 0 superscript 2 straight m squared straight g squared end fraction end root end style

Answered by Priyanka Kumbhar | 20th Jan, 2016, 04:51: PM