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I was reading about the Biot-Savart law and there were two formulas stated in the textbook. One was stated as the Biot-Savart law

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and the other was stated as the field due to a long, straight wire

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My question is what is and why is there difference between two formulas.

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The first formula is regarding the field created by a small arc length of a possibly-curved conductor. If we integrate that expression over the whole curve of the conductor, then we get the field at a point due to the conductor.

The second expression is the integrated version of the previous formula for an infinitely long straight wire. I wouldn't really consider the second to be the Biot-Savart law; rather it should be the field of a long straight conductor.

As an extra point, I'd like to say that it'd be much easier to find the field of a long straight wire conductor using Ampere's law rather than Biot-Savart.

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The first one is a specialized form to spacecurves. The second is even more specialized: its for line shaped conductor.

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  • $\begingroup$ An infinite line shaped conductor. $\endgroup$ – Not_Einstein Oct 18 '20 at 0:22

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