Lorrain & Corson: Help with Derivation of Fig 18.9/7.20

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The discussion focuses on deriving the substitution of dl with acos(phi)dphi in Lorrain & Corson's Figure 18.9 (or Figure 7.20 in the second edition). The key point is that dl has a magnitude of a dφ, and the derivation emphasizes the importance of the y-component of A, which incorporates the factor of cos(φ). Additionally, the text clarifies that the x-component of A is zero, as explained between equations 7.99 and 7.100. This understanding is crucial for accurately interpreting the equations in the context of the problem. The forum participants aim to clarify these derivations for better comprehension.
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can someone help me with a derivation in L&C. I am wrestling with Figure 18.9 (Fig 7.20 in second edition) . In Equations 18.36 and 18.37 (7.99 and 7.100), the element dl is substituted with acos(phi)dphi and I cannot see how this is derived.
 
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Welcome to Physics Forums! I have the 2nd edition of this book.

dl has a magnitude of a dφ. Moreover, we are only interested in the y-component of A, hence the additional factor of cos(φ). (The book explains, between equations 7.99 and 7.100, why the x-component of A must be zero.)
 
Many thanks Redbelly.
 

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