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Workdone

Asked by pottavarthiprashanthkumar | 11 May, 2021, 13:03: PM

Expert Answer

Workdone W to move the charge q from a point P to point Q is given as,

begin mathsize 14px style W space equals space minus q space integral subscript P superscript Q E times d l end style

First movement of -2μC charge is from A (0,5,0) to B ( 2,5,0) then to C(2,-1,0)

Workdone

begin mathsize 14px style W space equals space 2 cross times 10 to the power of negative 6 end exponent space open square brackets integral subscript 0 superscript 5 E subscript x space d x space plus space integral subscript 2 superscript negative 1 end superscript E subscript y d y close square brackets space end style

Workdone

begin mathsize 14px style W space equals space 2 space cross times space 10 to the power of negative 6 space space end exponent cross times space 10 cubed space open square brackets integral subscript 0 superscript 5 left parenthesis 3 x squared plus y right parenthesis space d x space plus space integral subscript 2 superscript negative 1 end superscript x space d y close square brackets space end style

Along the path AB , y = 5 . Along the path BC , x = 2 . Hence above integration becomes

begin mathsize 14px style W space equals space 2 space cross times space 10 to the power of negative 3 end exponent space open square brackets integral subscript 0 superscript 5 left parenthesis 3 x squared plus 5 right parenthesis space d x space plus space integral subscript 2 superscript negative 1 end superscript 2 space d y close square brackets space equals space 12 space m J end style

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If charge is moved from point A to B along the line y = 5 - 3x, then workdone w is given as

begin mathsize 14px style W space equals space 2 space cross times space 10 to the power of negative 6 end exponent space cross times integral subscript A superscript C E times d l space equals space 2 space cross times 10 to the power of negative 6 end exponent cross times integral subscript A superscript C left parenthesis space E subscript x space i with hat on top space plus space E subscript y space j with hat on top space right parenthesis space times left parenthesis d x space i with hat on top space plus space d y space j with hat on top right parenthesis space space equals space space 2 space cross times 10 to the power of negative 6 end exponent cross times integral subscript A superscript C space open parentheses E subscript x d x space plus space E subscript y d y close parentheses end style

begin mathsize 14px style W space equals space 2 space cross times space 10 to the power of negative 6 to the power of space end exponent cross times 10 cubed space open parentheses integral subscript 0 superscript 5 open parentheses 3 x squared plus y close parentheses d x space plus space integral subscript 2 superscript negative 1 end superscript x space d y close parentheses space end style

begin mathsize 14px style W space equals space 2 space cross times space 10 to the power of negative 3 to the power of space end exponent space open parentheses integral subscript 0 superscript 5 open parentheses 3 x squared space minus space 3 x space plus space 5 close parentheses d x space plus space 1 third integral subscript 2 superscript negative 1 end superscript left parenthesis 5 minus y right parenthesis space d y close parentheses space equals space 12 space m J end style

Answered by Thiyagarajan K | 11 May, 2021, 14:55: PM

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