Estimating Error in Meas. of Homog. Mag Field w/ Hall Effect Probe

[OGrowli]

Homework Statement

I'm trying to estimate the error in measurements of a homogenous magnetic field of known orientation using a Gauss-meter that uses a Hall-effect probe. The uncertainty in the measurements reported by the Gauss-meter are given the user's manual. Since the probe is held by hand, there will also be some uncertainty due to the orientation of the probe with respect to the magnetic field.
Let
$B_{hom}=$The homogeneous mag field
$B_{m}=$ The field value measured by the instrument
$\theta=$ The angle between $\vec{B}_{hom}$ and the direction normal to the surface of the hall element.

Homework Equations

The component of the magnetic field that gives rise to the Hall potential in the probe is perpendicular to the surface of the element, so,

$$B_{m}=B_{hom}\cos\theta\rightarrow B_{hom}= \frac{B_{m}}{\cos\theta}$$

The Attempt at a Solution

There is error in both $B_{m}$ and $\theta$. Through error propagation, the error in $B_{hom}$ will be,

$$\delta B_{hom}= \sqrt{ \left (\frac{\partial B_{hom}}{\partial B_{m}}\delta_{ B_{m}} \right )^{2} + \left ( \frac{\partial B_{hom}}{\partial \theta}\delta_\theta\right )^2}$$

$$\rightarrow \delta B_{hom}= B_{m}\sqrt{ \left ( \frac{\delta_{B_{m}}} {B_{M}} \right )^{2}+ \tan^{2}\left (\theta \right )\delta_{\theta}^{2}}$$

Here is where the ambiguity comes in, Theta is measured as zero by the person measuring the magnetic field. At zero, the term due to error in theta vanishes leaving only the term due to uncertainty in B_m. This is problematic. The formula gives no information about the error due to theta at the data point where I need the information. My question is how do I deal with this type of ambiguity? Taylor expansion maybe? Thank you for your replies.

 

[haruspex]

There are two answers to this. The first is that the equations are telling you that the final error is pretty insensitive to smallish errors in theta if theta is near 0. Beyond that, you may nevertheless want to estimate what happens for larger errors. For that, yes, you will need an expansion which keeps smaller terms, like cos(δθ) ≈ 1 - δθ²/2.

 

[OGrowli]

cool, thank you.