Electromagnetism - Stokes Theorem

The line integral of  u = yi − xj + zk around a circle of radius R in the xy-plane with center at the origin is equal to

A. 0
B. 2πR
C. 2πR²
D. πR²/4
E. 3R³

(GR9677 #43)

Solution:

Stokes' Theorem: The line integral of a vector field around a closed curve is equal to the surface integral of the curl over that vector field.

∮ F · dl = ∫ (∇ × F) · dA

Given:  u = yi − xj + zk 

∇ × u =

= 0 + 0 − k̂ − k̂ − 0 − 0 = − 2k̂

∫ (∇ × u) · dA = − ∫ 2k̂ dA = − 2k̂ ∫ dA  = − 2k̂ A  = − 2πR² k̂

Answer: C