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The charge of rod having length L is Q So Line charge density (lambda)=(Q)/(L) consider an element having length dy which is situated at distance if from origin.The charge dq=lambda dyNow the electric field due to une charge element isd vec(E)=(1)/(4piepsilon_(0))(dq)/(r^(2)) hat(r)nere vec(r)=x hat(i)-y hat(j)hat(r)=(x( hat(i))-y( hat(j)))/(sqrt(x^(2)+y^(2)))So d vec(E)=(1)/(4piepsilon_(0))(lambda dy)/(x^(2)+y^(2))(x( hat(i))-y( hat(j)))/(sqrt(x^(2)+y^(2)))=>d vec(E)=(1)/(4piepsilon_(0))(lambda dy)/((x^(2)+y^(2))^(3//2))(x hat(i)-y hat(j))So the total electric field due to rod is{:[ vec(E)=int_(0)^(L)(1)/(4piepsi_(0))(lambda dy)/((x^(2)+y^(2))//2)(x hat(i)-y hat(j))],[=> vec(E)=(1)/(4piepsilon_(0))int_(0)^(L)lambda[(xdy( hat(i)))/((x^(2)+y^(2))^(3//2))-(ydy( hat(j)))/((x^(2)+y^(2))^(3//2))]],[" Here put "g=x tan theta],[dy=xsec^(2)theta ... See the full answer