Module 1 : A Crash Course in Vectors
Lecture 5 : Curl of a Vector - Stoke's Theorem
  Example 23 :
A vector field is given by $\vec F(r,\theta,\phi) = f(r)\hat\phi$ where $\phi$ is the azimuthal angle variable of a spherical coordinate system. Calculate the line integral over a circle of radius $R$ in the x-y plane centered at the origin. Consider an open surface in the form of a hemispherical bowl in the northern hemisphere bounded by the circle.
  Solution :
  On the equatorial circle $\vec{dl} = R d\phi\hat\phi$. Hence,
 
\begin{displaymath}\oint \vec F\cdot\vec{dl} = \int_0^{2\pi} f(R) Rd\phi = 2\pi Rf(R)\end{displaymath}
  The expression for curl in spherical coordinates may be used to calculate the curl of $\vec F$. Since the field only has azimuthal component, the curl has radial and polar ( $\theta$) components.
 
\begin{eqnarray*} \nabla\times\vec F &=& \frac{1}{r\sin\theta}\frac{\partial}{\p... ...frac{f(r)}{r}+ \frac{\partial f(r)}{\partial r}\right)\hat\theta \end{eqnarray*}

         Back                                                                                                         Next