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    Hello all,

    I am trying to derive (for my understanding) the Biot-Savart law.

    Firstly, the derivation I'm following begins with the electrostatic E field result:

    \left | \vec{E} \right |=\frac{\mu _{0}q.\vec{r}}{4\pi r^{2}}

    but how does the constant here equal the standard SI Coulomb constant of:

    \frac{1}{4\pi \varepsilon _{0}} ?


    Also I still don't particularly understand the final stement of the Biot-Savart law, wrt the various vectors etc. I would appreciate a lucid 'translation' of what the Biot-Savart actually says.

    Here is the source of my confusion: http://study.com/academy/lesson/the-...-examples.html

    Thanks in advance.
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    (Original post by Sm0key)
    Hello all,

    I am trying to derive (for my understanding) the Biot-Savart law.

    Firstly, the derivation I'm following begins with the electrostatic E field result:

    \left | \vec{E} \right |=\frac{\mu _{0}q.\vec{r}}{4\pi r^{2}}

    but how does the constant here equal the standard SI Coulomb constant of:

    \frac{1}{4\pi \varepsilon _{0}} ?


    Also I still don't particularly understand the final stement of the Biot-Savart law, wrt the various vectors etc. I would appreciate a lucid 'transition' of what the Biot-Savart actually says.

    Here is the source of my confusion: http://study.com/academy/lesson/the-...-examples.html

    Thanks in advance.
    That "derivation" looks like a bunch of BS. Feynman derives the Biot-Savart law from the vector potential, A, whose existence itself comes from Maxwell's equations (\nabla \cdot B = 0 \Rightarrow B = \nabla \times A). You can see the details here, towards the end:

    http://www.feynmanlectures.caltech.edu/II_14.html

    So strictly, you need to know Maxwell's equations to get the Biot-Savart law in a mathematical way.

    However, historically, this law came from experimental evidence, and I guess it was based on the assumption that, if the E field follows an inverse square law, then so does the B field. It's tricky to show that directly experimentally though, since you would need to construct infinitesimally short wire segments - I don't know the details of how Biot/Savart came to the final result, but it must have needed some careful reasoning, as B fields produced by macroscopic wires *don't* follow inverse square laws (e.g. the field of a long wire goes as 1/r)
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    (Original post by atsruser)
    That "derivation" looks like a bunch of BS. Feynman derives the Biot-Savart law from the vector potential, A, whose existence itself comes from Maxwell's equations (\nabla \cdot B = 0 \Rightarrow B = \nabla \times A). You can see the details here, towards the end:

    http://www.feynmanlectures.caltech.edu/II_14.html

    So strictly, you need to know Maxwell's equations to get the Biot-Savart law in a mathematical way.

    However, historically, this law came from experimental evidence, and I guess it was based on the assumption that, if the E field follows an inverse square law, then so does the B field. It's tricky to show that directly experimentally though, since you would need to construct infinitesimally short wire segments - I don't know the details of how Biot/Savart came to the final result, but it must have needed some careful reasoning, as B fields produced by macroscopic wires *don't* follow inverse square laws (e.g. the field of a long wire goes as 1/r)
    Great thanks for that atsruser.

    I have had a flick through chapter 14 of Feynman's lectures and did infer that a good grasp of Maxwell's equations was a prerequisite (which is beyond the scope of my first year course). I've been going back over a couple of the relevant lecture slides, so that I can clarify what you're saying:

    Name:  Screen Shot 2016-04-30 at 11.16.45 pm.png
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    NB: Am I inserting this image correctly, how can I insert it inline at actual size instead of being thumbnail?

    Does the stated result come about from Maxwell's equations? In which case, given that I lack the mathematics (further vector calculus etc.) and physics to appreciate Maxwell's equations at present, should I just accept this first slide's statement for B field?

    Here's the second slide:
    Name:  Screen Shot 2016-04-30 at 11.17.50 pm.png
Views: 35
Size:  47.2 KB
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    (Original post by Sm0key)
    Great thanks for that atsruser.

    I have had a flick through chapter 14 of Feynman's lectures and did infer that a good grasp of Maxwell's equations was a prerequisite (which is beyond the scope of my first year course). I've been going back over a couple of the relevant lecture slides, so that I can clarify what you're saying:

    Name:  Screen Shot 2016-04-30 at 11.16.45 pm.png
Views: 42
Size:  54.3 KB

    Does the stated result come about from Maxwell's equations?
    I'm sorry for the late response - been busy.

    I believe that can indeed be derived from Maxwell's equations, and that it was first done by Heaviside. So I would not worry about knowing how to get that at the moment (I don't know myself, in fact).

    I have seen the Biot-Savart law derived starting from the B field for a moving point charge, by the way.

    In which case, given that I lack the mathematics (further vector calculus etc.) and physics to appreciate Maxwell's equations at present, should I just accept this first slide's statement for B field?
    Yes. However, be aware that if you have some persistence, you can learn a huge amount of this stuff on your own (vector calculus, Maxwell's equations, etc) simply by reading Vol II of the Feynman Lectures. It's a very, very thorough and detailed treatment of electromagnetism, and he develops lots of the maths as he's going along. In addition, his explanations (mostly) are crystal clear.

    I don't think Feynman actually derives that B field formula, however.

    From the p.o.v of the Biot-Savart law, you need to understand it (for most physics courses anyway) to the extent that you know what it's saying physically, and more importantly, that you can integrate it up given a certain wire geometry to find the B field due to the wire (e.g. for inifinite straight wires, current loops, solenoids and maybe a couple of other common ones).
 
 
 
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