Question

Consider a vortex filament of strength in the shape of a closed circular loop of radius R. Obtain an expression for the velocity induced at the center of the loop in terms of T and R

Answer 1

v = T/(2R)

Step-by-step explanation:

Given

R = radius

T = strength

From Biot - Savart Law

dv = (T/4π)* (dl x r)/r³

Velocity induced at center

v = ∫ (T/4π)* (dl x r)/r³

⇒ v = ∫ (T/4π)* (dl x R)/R³ (k) k: unit vector perpendicular to plane of loop

⇒ v = (T/4π)(1/R²) ∫ dl

If l ∈ (0, 2πR)

⇒ v = (T/4π)(1/R²)(2πR) (k) ⇒ v = T/(2R) (k)