In his famous experiment with electrons, J.J. Thomson measured the \"charge-to-m
ID: 1309691 • Letter: I
Question
In his famous experiment with electrons, J.J. Thomson measured the "charge-to-mass ratio" r = e/m, where e is the electron's charge and m its mass. A modern classroom version of this experiment finds the ratio r by accelerating electrons through a voltage V and then bending them in a magnetic field. The ratio r = emm is given by the formula In this equation, mu 0 is the permeability constant of the vacuum (equal to 4 pi times 10 - 7N/A2 exactly ) and N is the number of turns in the coil that produces the magnetic field; D is the diameter of the field coils, V is the voltage that accelerates the electrons, d is the diameter of the electrons' curved path, and I is the current in the field coils. A student makes the following measurements: N = 72 (exactly) D = 661 plusminus 2 mm V = 45.0 plusminus 0.2 volts d = 91.4 plusminus 0.5 mm I = 2.48 plusminus 0.04 amps Find the student's answer for the charge-to-mass ratio of the electron, with its uncertainty. [Assume all uncertainties are independent and random. Note that the first factor in (3.53) is known exactly and can thus be treated as a single known constant, K. The second factor is a product and quotient of four numbers, D2, V, d2, and I2, so the fractional uncertainty in the final answer is given by the rule (3.18). Remember that the fractional uncertainty in D2 is twice that in D, and so on.] (b) How well docs this answer agree with the accepted value r = 1.759 times 1011 C/kg? (Note that you don't actually need to understand the theory of this experiment to do the problem. Nor do you need to worry about the units; if you use SI units for all the input quantities, the answer automatically comes out in the units given.)Explanation / Answer
a.) r = 1.826 x 10^11 C/Kg (calculated by inserting all the values in the given formula)
dr/r = 2(dD/D) + dV/V + 2(dd/d) + 2(dI/I) = 2(2/661) + (0.2/45) + 2(0.5/91.4) + 2(0.04/2.48) = 0.053695
So dr= 0.053695*1.826 x 10^11 = 9.805 x 10^9
So r = 1.826 x 10^11 +/- 9.805 x 10^9
b.) How well means change in r divided by r
dr/r = (1.826 - 1.759)/1.759 = 0.038 = 3.8% error
So How well means = 96.2% (i.e 96.2% agrees the answer)