I have a question regarding finding the error associated with the $r_\mathrm e$ term in the equation

$$r_\mathrm e=\sqrt{\frac h{8\tilde B_\mathrm e\pi^2\tilde c\mu}}$$

Is there anyway I can find the uncertainty of $r_\mathrm e$ using any formula. $h$, $\pi$ and $\tilde c$ and $\mu$ all are constant. Only $\tilde B_\mathrm e$ have an uncertainty. Any readings that I could tap into?

  • $\begingroup$ Upto how much decimal you want accuracy? $\endgroup$ – JM97 Mar 27 '16 at 15:06
  • $\begingroup$ just 1 significant figure $\endgroup$ – Ong Zhi Qiang Mar 27 '16 at 15:06
  • $\begingroup$ I just want to know what is the general equation that I can tackle this type of question, there should be a formula that i could apply to determine the uncertainty $\endgroup$ – Ong Zhi Qiang Mar 27 '16 at 15:07
  • $\begingroup$ When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the multipliers. $\endgroup$ – JM97 Mar 27 '16 at 15:12

If the individual uncertainties of $h$, $\tilde B_\mathrm e$, $\pi$, $\tilde c$, and $\mu$ are not correlated, the total uncertainty of $r_\mathrm e$ may be estimated using the general formula:

$$u(r_\mathrm e) = \sqrt{\left(\frac{\partial r_\mathrm e}{\partial h}\right)^2 u^2(h)+\left(\frac{\partial r_\mathrm e}{\partial \tilde B_\mathrm e}\right)^2 u^2(\tilde B_\mathrm e)+\left(\frac{\partial r_\mathrm e}{\partial \pi}\right)^2 u^2(\pi)+\left(\frac{\partial r_\mathrm e}{\partial \tilde c}\right)^2 u^2(\tilde c)+\left(\frac{\partial r_\mathrm e}{\partial \mu}\right)^2 u^2(\mu)}$$

If only $\tilde B_\mathrm e$ has a relevant uncertainty, the formula can be simplified to:

$$u(r_\mathrm e) = \sqrt{\left(\frac{\partial r_\mathrm e}{\partial \tilde B_\mathrm e}\right)^2 u^2(\tilde B_\mathrm e)}$$

where the partial derivative of $r_\mathrm e$ with respect to $\tilde B_\mathrm e$ is

$$\frac{\partial r_\mathrm e}{\partial \tilde B_\mathrm e}= - \frac{1}{8}\frac{{\sqrt 2 h}}{{\sqrt {\frac{h}{{\tilde B_\mathrm e{\pi ^2}\tilde c\mu }}} {\tilde B_{\mathrm e}^2}{\pi ^2}\tilde c\mu }}$$


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