If $\displaystyle \bold{F}(x,y) = \frac{k(x \bold{i} + y \bold{j})}{x^{2}+y^{2}} $ find the work done by $\displaystyle \bold{F} $ in moving a unit charge along a straight line segment from $\displaystyle (1,0) $ to $\displaystyle (1,1) $.

So $\displaystyle \bold{F}(1,y) = \frac{k(\bold{i} + y \bold{j})}{1 + y^{2}} $. Then $\displaystyle x = 1, \ y = y $.

$\displaystyle k \int_{0}^{1} \frac{y}{1+y^{2}} \ dy $

$\displaystyle u = 1+y^{2} $

$\displaystyle du = 2y \ dy $

$\displaystyle \frac{k}{2} \int \frac{du}{u} $

$\displaystyle = \frac{k}{2} \int_{0}^{1} \ln|1+y^{2}| $

$\displaystyle = \frac{\ln 2}{2} $.

Is this correct?