Understanding Length and Area Elements for Electromagnetic Laws

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Finding length and area elements for Gauss's, Ampere's, and Biot-Savart Laws involves using infinitesimal distances, denoted as ds, which can be defined in Cartesian, cylindrical, or spherical coordinates. The choice of ds should align with the direction of the flux or current in the problem. It is important to understand that ds is a differential quantity and does not have a specific numerical value. Instead, one integrates with respect to s to solve the problems. Mastering these concepts for complex geometries will facilitate understanding simpler cases.
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I'm having a difficult time figuring out how to find length and area elements for Gauss's, Ampere's, and Biot-Savart Laws! Can someone please help explain! This applies to symmetrical, non-symmetrical, and infinite objects! I'm not asking what is the area of a sphere but, say with an infinite object, you use a small portion maybe called ds, well how do you figure how what your value of ds will be? And please don't give an example using a simple figure! That won't help me! I've noticed that if I understand the most difficult then I'm good to go for everything! Thanks :)
 
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ds is an infinitesimal distance which u can chose in cartesian coordinates, cylindrical or spherical coordinates. U want to chose ds along the line which contains the flux or the current, ...
 
You don't "figure out your value of ds". A differential has no value in terms of numbers. You integrate with respect to s.
 
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