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

Answer:

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)