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I am currently studying for O Level and I have been learning about electromagnetism for the past year. The topic has never clicked for me because I've never been taught the true relationship between electricity and magnetism. After looking online, I understand how electromagnetism and magnetism occur on their own (or at least enough to find some closure).

The thing that is still bugging is: why does the magnetic field around a straight current carrying wire 'rotate' a certain direction and not the other? To be specific, I want to understand its direction, not why it's circular. Is it just convention? I know it has something to do with cross-products but not how it applies here.

I know a lot of the topic is beyond my scope but I want to understand why this apparent 'asymmetry' occurs for the time being to rest my curiosity. Thanks for the any help.

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  • $\begingroup$ Sorry, I don't understand your question. What do you mean a magnetic field rotates a certain direction? $\endgroup$
    – PolaroidDreams
    Commented Apr 22, 2019 at 12:47
  • $\begingroup$ @expikx I mean the right-hand grip rule being used to determine the direction of the field. Say a current is coming towards you, according to the rule, the direction of the magnetic field should be anticlockwise -- but why? Why not clockwise? $\endgroup$
    – Typo
    Commented Apr 22, 2019 at 13:56
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    $\begingroup$ It can be clockwise. Anti-clockwise is just a convention similar to assigning positive angle for anti-clockwise rotation. You can define a whole lot of processes with the other convention. Physics stays the same. $\endgroup$
    – PolaroidDreams
    Commented Apr 22, 2019 at 13:59

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The “right-hand rule” for cross products is just a convention. Physics works fine if you adopt a “left-hand rule” instead. The direction of the magnetic field around a straight wire cannot be directly measured and is dependent on the right-hand convention. All that can be measured is how that field makes charged particles accelerate, and this involves two cross products (the other being in the Lorentz force), so it is independent of the convention.

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  • $\begingroup$ Just asking for some clarification: is the rule then used to just maintain consistency throughout physics? Is there actually nothing important about the direction of the field aside from the fact that it exists? $\endgroup$
    – Typo
    Commented Apr 22, 2019 at 14:01
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    $\begingroup$ Yes, the right-hand rule for cross products is just to maintain consistency throughout physics. The tangential nature of the magnetic field around a straight wire is significant. Whether it is considered to loop clockwise or counterclockwise around the wire is not significant and just a convention. You can thank or curse Josiah Gibbs for popularizing vector algebra and vector calculus and introducing the right-hand rule. I would bet that he was right-handed. $\endgroup$
    – G. Smith
    Commented Apr 22, 2019 at 18:01
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    $\begingroup$ There are other formalisms for electromagnetism, such as differential forms or Lorentz tensors, in which the magnetic field does not have the asymmetry that bothers you. $\endgroup$
    – G. Smith
    Commented Apr 22, 2019 at 18:01
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There is no asymmetry!

Or, the asymmetry is a result of using the magnetic vector $\mathbf b$, instead of the magnetic bivector $B$. The former is a directed line segment; the latter is a directed plane segment; they're perpendicular to each other.

$$\mathbf b=b_x\mathbf e_x+b_y\mathbf e_y+b_z\mathbf e_z$$ $$B=-b_x\mathbf e_y\mathbf e_z-b_y\mathbf e_z\mathbf e_x-b_z\mathbf e_x\mathbf e_y$$

Here $\mathbf e_x$ is the unit vector along the $x$-axis, and similarly for $y$ and $z$. And $\mathbf e_x\mathbf e_y$ is the unit bivector in the $x,y$-plane. (Actually there are two; the other is $\mathbf e_y\mathbf e_x=-\mathbf e_x\mathbf e_y$.) The vectors are being multiplied with the geometric product, which is associative, and satisfies $\mathbf v\mathbf v=\lVert\mathbf v\rVert^2$ for any vector $\mathbf v$. Multiplying a vector with a bivector produces two parts: another vector, called $\mathbf v\cdot B$; and a trivector, called $\mathbf v\wedge B$:

$$\mathbf vB=(v_x\mathbf e_x+v_y\mathbf e_y+v_z\mathbf e_z)(-b_x\mathbf e_y\mathbf e_z-b_y\mathbf e_z\mathbf e_x-b_z\mathbf e_x\mathbf e_y)$$ $$=(v_yb_z-v_zb_y)\mathbf e_x+(v_zb_x-v_xb_z)\mathbf e_y+(v_xb_y-v_yb_x)\mathbf e_z\\+(-v_xb_x-v_yb_y-v_zb_z)\mathbf e_x\mathbf e_y\mathbf e_z$$ $$\mathbf v\cdot B=(v_yb_z-v_zb_y)\mathbf e_x+(v_zb_x-v_xb_z)\mathbf e_y+(v_xb_y-v_yb_x)\mathbf e_z$$ $$=\mathbf v\times_3\mathbf b$$ $$\mathbf v\wedge B=(-v_xb_x-v_yb_y-v_zb_z)\mathbf e_x\mathbf e_y\mathbf e_z$$ $$=(-\mathbf v\cdot\mathbf b)I$$

Here $I=\mathbf e_x\mathbf e_y\mathbf e_z$ is the right-handed unit trivector (the left-handed one is $-I$), and $\times_3$ is the vector cross product which only works in 3D. (In contrast, geometric algebra works in any number of dimensions.) We also have $IB=\mathbf b$, an instance of Hodge duality.

So the Lorentz force $q\,\mathbf v\times_3\mathbf b$ can be replaced with $q\,\mathbf v\cdot B$. Geometrically, this is the projection of $\mathbf v$ onto the $B$ plane, rotated $90^\circ$ in that plane, and scaled by $q\lVert B\rVert$.

Here's a 3D illustration of the magnetic bivector field around a wire with current flowing downward. The magnitude of $B$ is represented by the area of the disk, which decreases with distance from the wire. The plane of $B$ always contains the wire. There is rotational symmetry, and mirror symmetry.

Wire magnetic field

More information: http://www.av8n.com/physics/clifford-intro.htm , https://en.wikipedia.org/wiki/Geometric_algebra

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  • $\begingroup$ While a lot of this is beyond me, I really appreciate this answer because I find geometric algebra fascinating. To anyone in the future reading, I really recommend watching this video on the topic: A Swift Introduction to Geometric Algebra (and its addendum). Probably the best math/physics video I've ever seen. $\endgroup$
    – Typo
    Commented Feb 14, 2023 at 8:29
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    $\begingroup$ From looking at that diagram, something tells me we should have either adopted the left-hand rule, or used the other direction for charge; that way you'd get the bivectors directed in the direction you'd expect from other physical phenomena (e.g. the direction a wheel would be spinning from water rushing by or how a moving gear rack would cause adjacent spur gears to move). $\endgroup$
    – Outis Nemo
    Commented Jul 8, 2023 at 19:26
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    $\begingroup$ @OutisNemo - Well, it is an observable fact that charge flowing downward causes nearby charge (of the same sign) to circulate in that way. If two charges are moving downward, they're attracted to each other. That's not what we'd expect from those analogies. $\endgroup$
    – mr_e_man
    Commented Jul 10, 2023 at 2:51
  • $\begingroup$ Well, I suppose that's true. I guess if you consider that it's charge of the opposite sign that starts circulating in that manner it makes a lot more physical sense, as opposite charges would be pulled in that direction as the first charge passes by. $\endgroup$
    – Outis Nemo
    Commented Jul 10, 2023 at 11:39
  • $\begingroup$ (I was ignoring the electrostatic force. Two charges of the same sign moving downward are magnetically attracted to each other. Or perhaps I should have referred to currents with no net charge.) $\endgroup$
    – mr_e_man
    Commented Jul 10, 2023 at 11:48

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