Magnetic Field Due to a Curved Wire Segment

In summary, the conversation discussed the use of limits of integration for an integral over the curved path AC. The text did not explicitly state the limits of integration, but the solution could be written as ##\int_S##, where S is the curve AC. The use of a line integral to denote summing the length elements over the path was also mentioned, but this symbol is typically used for integrating around a closed loop.
  • #1
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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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  • #2
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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  • #3
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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  • #4
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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  • #5
haruspex said:
No, that symbol is for integrating around a closed loop.
Oh, thank you for your help @haruspex !
 
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