Question

The velocity induced at a point \(P\) by the semi-infinite vortex filament is

• A. \(\dfrac{\Gamma}{4 \pi R}\)
• B. \(\dfrac{\Gamma}{2 \pi R}\)
• C. \(\dfrac{\Gamma}{\pi R}\)
• D. \(\dfrac{2 \Gamma}{\pi R}\)

Hint:

Use Biot–Savart law to evaluate induced velocities from vortex filaments.

Answer 1

The Correct Option is A

The velocity induced at a point \( P \) by a vortex filament can be determined using Biot-Savart law. For a semi-infinite vortex filament extending from \( P \) to infinity, the velocity induced at \( P \) is given by:

\[ v = \frac{\Gamma}{4 \pi R} \]

Here, \(\Gamma\) represents the circulation around the vortex filament and \( R \) is the perpendicular distance from the filament to the point \( P \). The formula \(\frac{\Gamma}{4 \pi R}\) describes the velocity field around a semi-infinite filament due to its geometry and the nature of induced velocities.

This option aligns with the correct understanding of vortex filament theory and matches the provided solution choice.