Relative uncertainty blows up near offset

Question

I'm looking to calculate the relative uncertainty for a magnetic field measurement. My device takes an initial reading but then performs operations on this using its calibration parameters. My equation is similar to this

$$B = a(x - x_0)$$

for calibration parameters $a$ and $x_0$ (the offset). Using the uncertainty formula

$$\Delta B = \sqrt{\sum{\left(\frac{\partial B(p_i)}{\partial p_i}\right)^2\Delta p_i^2}}$$

for calibration parameter $p_i$, we get

$$\Delta B^2 = (x-x_0)^2\Delta a^2 + a^2\Delta x_0^2$$

and a relative uncertainty

$$\left(\frac{\Delta B}{B}\right)^2 = \left(\frac{\Delta a}{a}\right)^2 + \left(\frac{\Delta x_0}{x-x_0}\right)^2$$

The relative uncertainty is what's bothering me. Again, my actual function is a little different but I am getting an asymptote near $x=x_0$ as I would expect with the equation given here. What's going on here? Unfortunately the device will need to take measurements near $x_0$ and this large rel. uncertainty is unwelcome.

Answer 1

There is nothing you can do! Since $B=0$ when $x=x_0$, the relative uncertainty (as you've calculated) diverges when $x\to x_0$ and is therefore undefined at that point. This is always the case for measuring a quantity that is zero. As a consequence, if you make measurements of the magnetic field near $x=x_0$, the relative error is going to be very large.