Magnetic Field Due to a Curved Wire Segment

AI Thread Summary
The discussion centers on the integration method used for calculating the magnetic field due to a curved wire segment. Participants question the necessity of specifying limits of integration, noting that the integral can yield the same result without them. It is clarified that the problem allows for defining the integration path as a curve, which can simplify the notation. Additionally, the use of line integrals is mentioned as a way to denote summing over the path without explicitly defining endpoints. The conversation concludes with an acknowledgment of the clarification provided.
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Homework Statement
Please see below
Relevant Equations
Please see below
For this problem,
1673674584161.png

The solution is,
1673674646262.png

However, why did they not use limits of integration for the integral in red? When I solved this, I used
1673674712595.png

as limits of integration.

I see that is not necessary since you get the same answer either way, but is there a deeper reason?

Many thanks!
 
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The text says "integrate over the curved path AC", so it was not essential to write that in the algebra. Also, one does not always have to specify the integration domain as a pair of endpoints. They could have defined S as the curve AC and written ##\int_S##.
 
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haruspex said:
The text says "integrate over the curved path AC", so it was not essential to write that in the algebra. Also, one does not always have to specify the integration domain as a pair of endpoints. They could have defined S as the curve AC and written ##\int_S##.
Thanks for your help @haruspex ! That second notation you mention makes more sense than their single integral over ds. I think another way to avoid implicitly defining an integration domain is by using a line integral
1673676771265.png
to denote that we are summing the length elements over the path, correct?
 

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Callumnc1 said:
Thanks for your help @haruspex ! That second notation you mention makes more sense than their single integral over ds. I think another way to avoid implicitly defining an integration domain is by using a line integral View attachment 320352to denote that we are summing the length elements over the path, correct?
No, that symbol is for integrating around a closed loop.
 
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haruspex said:
No, that symbol is for integrating around a closed loop.
Oh, thank you for your help @haruspex !
 
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