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Asked by Vidushi412 | 04 Apr, 2019, 11:50: AM
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Let us assume a uniformly charged sphere of radius R and total charge Q. Charge density ρ = Q/(4πR3)
 
if a charge -q is moving from a surface point towards centre, workdone W is given by,
 
W = begin mathsize 12px style integral subscript R superscript 0 F d r space equals negative space integral subscript R superscript 0 space q E left parenthesis r right parenthesis d r space equals space minus integral subscript R superscript 0 space q space fraction numerator rho cross times 4 πr cubed over denominator 4 πε subscript straight o straight r squared end fraction d r space equals space fraction numerator q rho over denominator straight epsilon subscript straight o end fraction integral subscript 0 superscript R r space d r space space equals space fraction numerator q rho over denominator straight epsilon subscript straight o end fraction R squared over 2 end style ............................(1)
workdone as given by eqn.(1) will be equal to maximum kinetic energy (1/2)kR2 at equilibrium position O, where k is force constant
and it is given by k = mω2 , where m is mass of charge and ω is angular frequency
 
( maximum kinetic energy at equilibrium position for a SHM is (1/2)kA2 , where A is amplitude.
Here we have assumed maximum amplitude equals Radius of sphere)
 
begin mathsize 12px style 1 half m omega squared space R squared space equals space fraction numerator q rho over denominator epsilon subscript o end fraction R squared over 2 omega squared space equals space fraction numerator q rho over denominator m space epsilon subscript o end fraction space equals space fraction numerator q over denominator m space epsilon subscript o end fraction space fraction numerator Q over denominator 4 πR cubed end fraction omega space equals space square root of fraction numerator 1 over denominator 4 straight pi straight epsilon subscript straight o end fraction fraction numerator q Q over denominator m R cubed end fraction end root end style
Answered by Thiyagarajan K | 04 Apr, 2019, 12:57: PM

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