- #1
tronter
- 185
- 1
If [tex] \bold{F}(x,y) = \frac{k(x \bold{i} + y \bold{j})}{x^{2}+y^{2}} [/tex] find the work done by [tex] \bold{F} [/tex] in moving a unit charge along a straight line segment from [tex] (1,0) [/tex] to [tex] (1,1) [/tex].
So [tex] \bold{F}(1,y) = \frac{k(\bold{i} + y \bold{j})}{1 + y^{2}} [/tex]. Then [tex] x = 1, \ y = y [/tex].
[tex] k \int_{0}^{1} \frac{y}{1+y^{2}} \ dy [/tex]
[tex] u = 1+y^{2} [/tex]
[tex] du = 2y \ dy [/tex]
[tex] \frac{k}{2} \int \frac{du}{u} [/tex]
[tex] = \frac{k}{2} \int_{0}^{1} \ln|1+y^{2}| [/tex]
[tex] = \frac{k\ln 2}{2} [/tex].
Is this correct?
So [tex] \bold{F}(1,y) = \frac{k(\bold{i} + y \bold{j})}{1 + y^{2}} [/tex]. Then [tex] x = 1, \ y = y [/tex].
[tex] k \int_{0}^{1} \frac{y}{1+y^{2}} \ dy [/tex]
[tex] u = 1+y^{2} [/tex]
[tex] du = 2y \ dy [/tex]
[tex] \frac{k}{2} \int \frac{du}{u} [/tex]
[tex] = \frac{k}{2} \int_{0}^{1} \ln|1+y^{2}| [/tex]
[tex] = \frac{k\ln 2}{2} [/tex].
Is this correct?
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