Physics307L F09: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

General formula, uncorrelated uncertainties (From wikipedia)

Given $X=f(A, B, C, \dots)$

$\sigma_X^2=\left (\frac{\partial f}{\partial A}\sigma_A\right )^2+\left (\frac{\partial f}{\partial B}\sigma_B\right )^2+\left (\frac{\partial f}{\partial C}\sigma_C\right )^2+\cdots$

Specific example, Millikan oil drop

Formulae for droplet charge (from John Callow):

$q = \left[400{\pi}d\left(\frac{1}{g{\rho}}{\left[\frac{9*{\eta}}{2}\right]^3}\right)^{\frac{1}{2}}\right]*\left[\left(\frac{1}{1+\frac{b}{pa}}\right)^{\frac{3}{2}}\right]*\left[\frac{V_f+V_r\sqrt {V_f}}{V}\right] e.s.u.$

$a = \sqrt {\left(\frac{b}{2p}\right)^2 + \frac{9{\eta}*V_f}{2g{\rho}}}- \left(\frac{b}{2p}\right)$

Vf = fall velocity
Vr = rise velocity. These (along with others) can have comparable effects on q.