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Question

A simple pendulum with bob of mass m and conducting wire of length L swings under gravity through an angle 2 theta. The earth’s magnetic field component in the direction perpendicular to swing is B. Maximum potential difference induced across the pendulum is

  1. 2 B L sin invisible function application open parentheses fraction numerator theta over denominator 2 end fraction close parentheses left parenthesis g L right parenthesis to the power of 1 divided by 2 end exponent    
  2. B L sin invisible function application open parentheses fraction numerator theta over denominator 2 end fraction close parentheses left parenthesis g L right parenthesis    
  3. B L sin invisible function application open parentheses fraction numerator theta over denominator 2 end fraction close parentheses left parenthesis g L right parenthesis to the power of 3 divided by 2 end exponent    
  4. B L sin invisible function application open parentheses fraction numerator theta over denominator 2 end fraction close parentheses left parenthesis g L right parenthesis to the power of 2 end exponent    

The correct answer is: 2 B L sin invisible function application open parentheses fraction numerator theta over denominator 2 end fraction close parentheses left parenthesis g L right parenthesis to the power of 1 divided by 2 end exponent



    Þ h equals L left parenthesis 1 minus cos invisible function application theta right parenthesis (i)
    Maximum velocity at equilibrium is given by
    \ v to the power of 2 end exponent equals 2 g h equals 2 g L left parenthesis 1 minus cos invisible function application theta right parenthesis equals 2 g   L open parentheses 2 sin to the power of 2 end exponent invisible function application fraction numerator theta over denominator 2 end fraction close parentheses
    Þ v equals 2 square root of g L end root sin invisible function application fraction numerator theta over denominator 2 end fraction
    Thus, max. potential difference
    V subscript m a x end subscript equals B cross times 2 square root of g L end root sin invisible function application fraction numerator theta over denominator 2 end fraction L equals 2 B L sin invisible function application fraction numerator theta over denominator 2 end fraction left parenthesis g L right parenthesis to the power of fraction numerator 1 over denominator 2 end fraction end exponent.

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