# Relative uncertainty blows up near offset

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.

• Why are you concerned with the relative error for a small quantity to begin with? What you need to be concerned about is the absolute error, which can be nulled out (remove the magnetic field or remove the measurement device from the magnetic field). Jun 25, 2015 at 4:43

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.