2 Consider the same vortex filament as in Problem 5.1. Consider also a straight line through the center of the loop, perpendicular to the plane of the loop. Let A be the distance along this line, measured from the plane of the loop. Obtain an expression for the velocity at distance A on the line, as induced by the vortex filament.

The expression for the velocity at distance A on the line is

Explanation:

The free body diagram of the circular voltage filament is shown on the first uploaded image

Looking at the diagram we see that a straight pass through the center of the loop and this line is perpendicular to the plane of the loop

    R is the radius of this vortex filament ,   denoted the strength of the vortex filament , V is  the velocity that is been induce due to the distance A traveled, is the elemental length of the vortex filament

   Now the velocity that is been induced perpendicular to the plane of the loop  According to Biot-Sarvart law is  mathematically represented as

Now the velocity induced at the distance A on the line is mathematically represented as

This is  because   from the diagram applying Pythagoras theorem  

This is  because    from the diagram applying SOHCAHTOA