Force F has units of kg m/s2 . The electric force between an object of
charge Q and a second object of charge q separated by a distance r is given by F= k Qq/ r2 , where k is
a constant. In order to determine the charge of q, the electric force between the two charges is
measured to accuracy ÄF, Q is known to a accuracy of ÄQ, and r is known to an accuracy Är.
Assuming that all these errors are uncorrelated, what is the fractional error in the determination of q
(Fractional error = is Äq/q ) . Start with q= F r2/ (kQ)
I find it convenient to put the expression in logarithm form first.
lnq = lnF - lnk -lnQ +2*lnr
Take the differential due to small errors in F, Q and r:
dq/q = dF/F -dQ/Q +2 dr/r
(There is no appreciable uncertainty in k, so dk = 0)
When comparing the effects of uncorrelated separate error sources, you can ignore the + or - sign, since the error of each type goes randomly both ways.
The fractional error is dq/q is the square root of the sum of
(dF/F)^2 + (dQ/Q) + 2(dr/r)^2^2
That is the way random uncorrelated errors add.
I am not able to read the error numbers you have provided; just use the formula above.
To find the fractional error in the determination of q, you can start with the equation q = Fr^2 / (kQ).
1. Let's differentiate both sides of the equation with respect to q:
dq = d(Fr^2 / (kQ))
2. Next, let's divide both sides by q:
dq/q = d(Fr^2 / (kQ)) / (Fr^2 / (kQ))
3. To simplify the expression, let's cancel out the common factors:
dq/q = d(Fr^2) / (Fr^2) = (2dF/F) + (dr/r)
4. Now, let's find the fractional error (dF/F) and (dr/r) in terms of their respective absolute errors (ΔF, Δr) and values (F, r):
dF/F = ΔF / F
dr/r = Δr / r
5. Substituting these expressions back into the equation, we get:
dq/q = (2ΔF/F) + (Δr/r)
Hence, the fractional error in the determination of q is given by (2ΔF/F) + (Δr/r).