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

v = T / (2R)

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)