Question

A thin semi–circular ring of radius r has a positive charge q distributed uniformly over it. The net field $straight E with rightwards arrow on top$ at the center O is

$fraction numerator straight q over denominator 4 straight pi squared straight epsilon subscript 0 straight r squared end fraction space bold j with hat on top$
$negative fraction numerator straight q over denominator 4 straight pi squared straight epsilon subscript 0 straight r squared end fraction$
$negative fraction numerator straight q over denominator 2 straight pi squared straight epsilon subscript 0 straight r squared end fraction straight j with hat on top$
$fraction numerator straight q over denominator 2 straight pi squared straight epsilon subscript 0 straight r squared end fraction bold space bold j with bold hat on top$

Solution

negative fraction numerator straight q over denominator 2 straight pi squared straight epsilon subscript 0 straight r squared end fraction straight j with hat on top

straight E space equals space fraction numerator 1 over denominator 4 πε subscript 0 end fraction straight q over straight r squared fraction numerator sin space straight pi divided by 2 over denominator straight pi divided by 2 end fraction

space equals space fraction numerator 1 over denominator 4 πε subscript 0 end fraction straight q over straight r squared 2 over straight pi

straight E with rightwards arrow on top space equals space minus space fraction numerator straight q over denominator 2 straight pi squared straight epsilon subscript 0 straight r squared end fraction straight j with hat on top

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