By J. E. Parton, S. J. T. Owen, M. S. Raven (auth.)

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V x F) may be compared with the scalar triple product P. (Q x R) which yields zero if two of the vectors are the same. 1 The laplacian The laplacian or lap operator in the rectangular cartesian coordinate system is a2 jax 2 + a2 foy 2 + o2 /oz 2 ' and is written as 'V2 . It can operate on scalar and vector fields and 'V 2 S may be defined as the divergence of the gradient of the scalar field 18 APPLIED ELECTROMAGNETICS \7 2 F may be defmed as the gradient of the divergence ofF minus the curl of the curl ofF and in the rectangular coordinate system is equal to the vector sum of the laplacian operations on the three scalar components of the field vector F.

In order to illustrate the principle of curl, H. H. Skilling uses the concept of a paddle wheel in a river, in which the velocity of flow decreases with depth. 13, and its axis when rotating with maximum velocity would give the direction of curl. 13 Illustration of cur/. fieldS by means of its gradient or else the line integral around a closed path in the corresponding vector field would have been zero. Although curl suggests that motion or flow in curves is present this is not strictly necessary, as in the example quoted.

If we have a choice of axes these can be ftxed so that the charged surface lies in the yz plane. The point P can be on the x axis, that is coordinates (a, 0, 0). 5b. 5 (a) Surface in yz plane. (b) Infinitely long plane. The surface element can be an infinite strip dy wide having PL = Ps dy. From the previous section dE= Ps dy a 2rt€oP P in whichy =a cot {3; dy =-a cpsec 2 {3 d{3 and p =a cosec {3. £§__ 2ne 0 J f3. \ En ~ I I ~ I I ~ I I~ I (a) = lia. 6 Field due to uniform surface charge Ps coulombs per square metre.