FOR REFERENCE : Equate the approximations in Steps 7 and 8, and let the height o

Question

FOR REFERENCE :

Equate the approximations in Steps 7 and 8, and let the height of the rectangle delta z approach zero to show sigma E / sigma z = sigma B / sigma t We now work with Faraday's Law (3) and determine its implications. Imagine a small rectangular surfaces in the xz-plane with one vertex at (0, 0, z); it has length a and height delta z (Figure 4). Let C be the boundary of S. Notice that two sides of C are orthogonal to E and make no contribution to the integral on the left side of (3). Integrating around C in the direction shown in the figure, show that Considering the right side of (3) show that

Explanation / Answer

The approx in #7 is just   (delta E) a , and the approx in #8 is    (- del B/del t) a (delta z)    , where the word "del" is the partial derivative operator and symbol .

Equating these two approximations, we see that "a" cancels out.   

Dividing both sides by (delta z) gives:    (delta E) / (delta z) = - (del B . del t) .   

Taking the limit of the above expression as delta z goes to zero gives the desired result that:     (del E/ del z) = - (del B / del t ).

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