The Curl of a Vector Field and Stokes

The Curl of a Vector Field and Stokes - with a nonzero curl...

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The Curl of a Vector Field and Stokes' Theorem The second major concept from vector calculus that applies to magnetic fields is that of the curl of a vector function. Take again our vector field F = ( P , Q , R ) . The curl of this vector field is defined as: = - , - , - Clearly this equation is a bit more complicated, but it gives us a lot more information. The curl, unlike the divergence, is itself a vector field, defined by a single vector at each point. Physically speaking, curl measures the rotational motion of a vector field. Again using our water analogy, a nonzero curl indicates an eddy or a whirlpool. At a given point in the field, the curl at that point tells us the axis of rotation of the field about that point. If the curl is zero, there is no axis of rotation, and thus no circular motion. Unlike magnetic fields, electric fields never have curls. Recall that the line integral over any closed loop in an electric field is zero, implying that the field cannot "curve" around, as a field
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Unformatted text preview: with a nonzero curl would. Just as Gauss' Theorem relates surface integrals and volume integrals using divergence, Stokes' Theorem relates surface integrals and line integrals using curl. Given a closed curve that encompasses a surface, · ds = · da where the left side is a line integral and the right side is a surface integral. Again, we pay special attention to the special case in which the curl is zero. In this case, the integral of the field around any closed loop is zero. Electric fields have this property. It is incredibly important when using Gauss' and Stokes' Theorem to remember that one must deal with closed surfaces (with Gauss' Theorem) or closed loops (with Stokes' Theorem). Otherwise the equations are not applicable. Having established the concepts of curl and divergence, and related them to vector calculus through our two theorems, we can apply these concepts to magnetic fields....
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This note was uploaded on 02/09/2012 for the course PHY PHY2053 taught by Professor Davidjudd during the Fall '10 term at Broward College.

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