Physics307L:Schedule/Week 13 agenda/Error
General question
- Recall when q = f(x) and we know x_best and sigma_x
- How about when q = f(x,y, ...) and uncertainties in more than one x, y are important?
- Let's derive
Key points after our derivation
- General formula is an approximation (small relative uncertainties)
- Usually written down to ignore covariance
Given [math]\displaystyle{ X=f(A, B, C, \dots) }[/math]
[math]\displaystyle{ \sigma_X^2=\left (\frac{\partial f}{\partial A}\sigma_A\right )^2+\left (\frac{\partial f}{\partial B}\sigma_B\right )^2+\left (\frac{\partial f}{\partial C}\sigma_C\right )^2+\cdots }[/math]
example formulas
Specific example, Millikan oil drop
Formulae for droplet charge (from John Callow):
[math]\displaystyle{ q = \left[400{\pi}d\left(\frac{1}{g{\rho}}{\left[\frac{9*{\eta}}{2}\right]^3}\right)^{\frac{1}{2}}\right]*\left[\left(\frac{1}{1+\frac{b}{pa}}\right)^{\frac{3}{2}}\right]*\left[\frac{V_f+V_r\sqrt {V_f}}{V}\right] e.s.u. }[/math]
[math]\displaystyle{ a = \sqrt {\left(\frac{b}{2p}\right)^2 + \frac{9{\eta}*V_f}{2g{\rho}}}- \left(\frac{b}{2p}\right) }[/math]
[math]\displaystyle{ V_f }[/math] = fall velocity
[math]\displaystyle{ V_r }[/math] = rise velocity. These (along with others) can have comparable effects on q.
