Answer:
[tex]\vec{V} = \frac{\Gamma}{2R}\vec{A}[/tex]
Explanation:
We define our values according to the text,
R= Radius
[tex]\vec{V} =[/tex]Velocity
[tex]\Gamma =[/tex]Strenght of the vortex filament
From this and in a vectorial way we express an elemental lenght of this filmaent as [tex]\vec{dl}[/tex]. So,
[tex]\vec{dl}x\vec{r} = R*dl*\vec{A}[/tex]
Where [tex]\vec{A}[/tex] imply a vector acting perpendicular to both vectors.
Applying Biot-Savart law, we have,
[tex]\vec{V} =\frac{\Gamma}{4\pi}\int\frac{\vec{dl}x\vec{r}}{r^3}[/tex]
Substituting the preoviusly equation obtained,
[tex]\vec{V} = \frac{\Gamma}{4\pi}\int\frac{R*dl*\vec{A}}{R^3}[/tex]
[tex]\vec{V} = \frac{\Gamma}{4\pi R^2}\int^{2\pi R}_0 dl*\vec{A}[/tex]
[tex]\vec{V} = \frac{\Gamma(2\pi R \vec{A})}{4\pi R^2}[/tex]
So we can express the velocity induced is,
[tex]\vec{V} = \frac{\Gamma}{2R}\vec{A}[/tex]
