LH vs Gaussian units in Biot-Savart Law

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Discussion Overview

The discussion centers on the relationship between the Biot-Savart law in Gaussian and Lorentz-Heaviside (LH) units, specifically addressing the presence of different factors in the formulas used in each unit system. Participants explore the implications of these factors on the transformation of magnetic field strength and current.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the presence of a ##\frac{1}{\sqrt{4\pi}}## factor in the LH units version of the Biot-Savart law, noting the difference from Gaussian units.
  • Another participant reiterates the transformation relationship between the magnetic field in LH and Gaussian units, suggesting that the extra factor arises from the transformation of the magnetic field strength.
  • A third participant introduces the relationship between current and charge, arguing that the transformation for current would introduce an additional ##\sqrt{4\pi}## factor into the numerator, complicating the resolution of the issue.
  • One participant challenges the previous claims by suggesting that the mathematical transformations should be consistent and that the factor in question is indeed correct.

Areas of Agreement / Disagreement

Participants express differing views on the correct interpretation of the transformation factors between the two unit systems, indicating that the discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Participants have not reached a consensus on the mathematical transformations involved, and there are indications of potential misunderstandings regarding the application of unit transformations to the components of the Biot-Savart law.

avikarto
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I have a question regarding the relationship between the Biot-Savart formula in Gaussian and Lorentz-Heaviside units. In Gaussian, we have a ##\frac{1}{c}## outside the integral, but in LH units we have a ##\frac{1}{4\pi}\frac{1}{c}##. This does not make sense, considering the transformation between Gaussian and LH for B is that ##B_{LH}=\frac{B_G}{\sqrt{4\pi}}##. Where does the extra ##\sqrt{4\pi}## come from? Thanks.

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avikarto said:
the transformation between Gaussian and LH for B is that ##B_{LH}=\frac{B_G}{\sqrt{4\pi}}##. Where does the extra ##\sqrt{4\pi}## come from? Thanks.

Source
it comes from I.
 
If we take ##I=q v##, then using that ##q_{LH}=\sqrt{4\pi} q_G##, we would have that ##I_{LH}=\sqrt{4\pi} I_G##. This seems to brings an extra ##\sqrt{4\pi}## into the numerator, not the denominator, and would therefore remove the factor all together. I don't understand how this resolves the issue. Besides, the transformation on B should be the transformation on B, without also having to transform its subcomponents additionally. Could you please elaborate?
 
avikarto said:
This seems to brings an extra 4π−−√4π\sqrt{4\pi} into the numerator, not the denominator, and would therefore remove the factor all together.
Check your math. Write the equation in one set of units, and then substitute. The factor is correct.
 

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