Simple Nature - Light and Matter

i.e.,F = axˆx + byˆx + cˆx + dxŷ + eyŷ + fŷ .The only terms whose curls we haven’t yet explicitly computed arethe a, e, and f terms, and their curls turn out to be zero (homeworkproblem 50). Only the b and d terms have nonvanishing curls. Thecurl of this field iscurl F = curl (byˆx) + curl (dxŷ)= b curl (yˆx) + d curl (xŷ) [scaling]= b(−ẑ) + d(ẑ) [found previously]= (d − b)ẑ .i / A cyclic permutation of x,y, and z.j / Example 17.But any field in the x − y plane can be approximated with thistype of field, as long as we only need to get a good approximationwithin a small region. The infinitesimal Ampèrian surface occurringin the definition of the curl is tiny enough to fit in a pretty small region,so we can get away with this here. The d and b coefficients canthen be associated with the partial derivatives ∂F y /∂x and ∂F x /∂y.We therefore have( ∂Fycurl F =∂x − ∂F )xẑ∂yfor any field in the x − y plane. In three dimensions, we just need togenerate two more equations like this by doing a cyclic permutationof the variables x, y, and z:(curl F) x = ∂F z∂y − ∂F y∂z(curl F) y = ∂F x∂z − ∂F z∂x(curl F) z = ∂F y∂x − ∂F x∂yA sine wave example 17⊲ Find the curl of the following electric fieldand interpret the result.E = (sin x)ŷ ,⊲ The only nonvanishing partial derivative occurring in this curl is(curl E) z = ∂E y∂x= cos x ,682 Chapter 11 Electromagnetism