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Today we started learning about Biot-Savart law. While deriving the formula, our professor listed out a bunch of relations, which have been experimentally determined,

->dB is directly proportional to current, length of element, and inversely proportional to square of distance.

This all seemed ok and relatable, and then he wrote another proportionality-> dB is directly proportional to sine of angle between position vector and current flow.

This seemed completely non intuitive to me. Is there any logical way of deducing this?

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    $\begingroup$ Have you learned about the cross product between two vectors? $\endgroup$
    – Michael Seifert
    Commented May 13, 2020 at 15:46
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    $\begingroup$ I do not understand what you mean when you ask for a logical deduction of facts presented to you as experimentally determined. Typically, such "experimental facts" serve as the input to a theory from which further things are then deduced. From what other facts do you want this to be deduced? $\endgroup$
    – ACuriousMind ♦
    Commented May 13, 2020 at 16:11
  • $\begingroup$ @MichaelSeifert I do know....you may tell it just comes from the cross product of dl and I but during the derivation they were found individually $\endgroup$
    – DatBoi
    Commented May 13, 2020 at 16:17
  • $\begingroup$ @ACuriousMind I have seen a few places where it's been mentioned like this has something to do with relativity, and thus out of the scope of syllabus $\endgroup$
    – DatBoi
    Commented May 13, 2020 at 16:18
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    $\begingroup$ Well, just like the aforementioned proportional relationships, isn’t ”$dB$ is directly proportional to sine of angle…” also something that can be experimentally observed? $\endgroup$
    – Jonathan Huang
    Commented Jan 3 at 7:29

2 Answers 2

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The Biot-Savart law is derived from experiment. It cannot be obtained by logic alone, except from other theoretical formulations that also derive from experiment. Nobody had any 'logical' intuition for magnetic fields generated by currents before they were discovered by Oersted in 1820, preceded by Romagnosi in 1802, and even Oersted did not have such an intuition and he was unable to give a theoretical explanation nor a mathematical model. https://en.wikipedia.org/wiki/Gian_Domenico_Romagnosi

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For this problem, I will think from the mathematical point of view. Magnetic field, current flow and position vector are all vector quantities. Hence, mathematically, it must be done in "vectorial" way. Vectors multiplication that result a vector is cross product. And mathematically, cross product is proportional to the $sine$ of the angle between the two vectors multiplied.

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  • $\begingroup$ Hmm mathematically it seems ok but how do you explain it practically? $\endgroup$
    – DatBoi
    Commented May 14, 2020 at 6:34

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