Physics 1c Prac -- EXPERIMENT 9

This is an annotated version of section 4 of Experiment 9. The point of section 4 is to increase your sophistication in dealing quantitatively with errors and uncertainties as they enter the design of experiments and the comparison of expectation and observation. If, after careful analysis, expectations and observations agree within an anticipated, understood range, then one has some confidence that the theory and experiment are reasonable. If not, it's a red flag. Also, a careful error analysis will point to which aspect of the experiment (or theory) would yield the most cost-effective improvement. But first, a couple of construction notes:

• When winding the second coil, don't discard magnet wire just because the coils have different numbers of turns. Your goal is to maximize the force between the coils for your total, given amount of wire.

• Most people have greater success with a razor blade than a soldering iron in removing the magnet wire insulation.

• The sketch on page 67 is a side view. The wires feeding the upper and lower coils should be long enough to allow free movement.

``4. Balance Sensitivity

Try using two or three of the large $\frac 14$-20 nuts and one of the smaller 10-32 nuts as weights on the balance arm. Your instructor will tell you what their measured masses are."

Current estimates are

$m_{\frac 14-20}=3.12\pm 0.05$ g

$m_{10-24}=1.36\pm 0.04$ g

The most recent smaller nuts are 10-24's rather than 10-32's. The substantial uncertainty in the nut masses comes from nut-to-nut variation in the (crude) manufacturing process. You probably could get a better determination of your own set if you have access to a good scale, e.g., in someone's lab.

``Use nuts opposite the coil to generate almost enough torque to lift the balance arm off the Styrofoam. Then move the small nut (leaving the others alone to avoid creating extra uncertainty) to just lift the arm off. Measure the position of this weight at liftoff for five or more successive trials. Look at the spread of your data..."

It is worth thinking about why you don't get the same result each time. (For example, do you really do exactly the same thing each time? What effects are and what effects are not included in your ``theory'' of the balance -- even before the current flows? Why are approximations and simplifications ALWAYS necessary in physics?)

``...and make a common sense estimate of how accurately you will know the torque for a single trial."

Recall the basic description of this experiment, e.g., as given in the first paragraph on page 68. A very sensitive measurement is to be performed using a mechanical analog of something you did in Experiments 2, 5, and 6. In particular, a large, uninteresting quantity is balanced out, and you look only at a small change -- in this case the difference between no current and current. The point of first measuring several (``at least five'') lift-offs is to see how much the small-nut position typically varies to counterbalance what is supposedly the same torque from the other side of the fulcrum. You should convert this spread of the small-nut position (calculated as described below) into a corresponding uncertainty in torque and compare it to the anticipated extra torque due to the attraction of the currents. (Note that you neither know nor care about the actual total torque with the currents off.)

People typically assume that the average of a series of measurements should be taken to represent one's best estimate of the ``actual, true'' value. (The extent to which this is or is not an appropriate strategy is a fascinating but very deep question.) ZAP! suggests that you compute the average of the magnitudes of the differences between the individual measurements and this ``true'' value. (``Standard deviation'' is defined as the RMS value of these differences rather than the average of absolute values; it satisfies some nicer theorems in formal probability theory, but either would do for the purpose at hand, which is to estimate the typical difference between a single measurement and the desired quantity.)

``Question: Calculate the force between your coils for a current of 1 ampere, referring to the formula you derived in the background section. Compare the torque due to this force with the uncertainty in torque you found just above, and estimate the expected percent accuracy of your experiment, assuming that this is the only source of error."

This calculation sheds light on the question of whether your balance is sensitive enough to measure the force between two 1 amp currents -- worth estimating before actually attempting the measurement.

``Think of other sources of error, such as the uncertainty in coil separation, and write them down in a table, with estimates of their size..."

You will ultimately want to compare your expectation to your observation of the force between the currents. Many pieces go into the determinations of each of these numbers. There are measurements that you make, measurements that come from other sources, and formulas (or theories) relating the various quantities. With each of these pieces there is an associated uncertainty. Some of these uncertainties are of the type most evident in your small-nut position measurements. In particular, we imagine that the average of a finite number of trials gets closer and closer to the actual ``true'' value as the number of trials increases. If, indeed, the individual errors are ``random'', then for $N$ trials the uncertainty in the average is generally believed to be $1/\sqrt{N}$ times the uncertainty in an individual trial. Errors that behave in this fashion are termed ``statistical.''

However, there are other kinds of errors as well. Besides outright boo-boos, one can have uncertainties that do not diminish no matter how many times the measurement is repeated. For example, your apparatus, e.g., your ruler or multimeter scale, may have limited resolution. Most people simply cannot do better than read a graduated scale to better than 1/10 of one marked division. Repetition can never beat this limitation. Also, a given instrument may have an inherent inaccuracy, be it fixed or erratic. (An estimate of this might be provided by the manufacturer, or you might have to figure it out for yourself. For example, how good are the markings on rulers? Are the lines where they are supposed to be, e.g., within a line thickness, or are they a bit off? How would you tell? -- You probably get what you pay for!)

There are errors or uncertainties that one introduces quite consciously. It may be that at a given point in the investigation it is appropriate to ignore certain effects rather than deal with them explicitly. Generally, this would be because they are believed to be, in some sense, ``small''. They may be things that one could, with greater effort, account for explicitly. Or they may be things that one can only estimate crudely. Here is a very relevant example:

The force per unit length between two straight, parallel, current carrying wires at separation $d$ is purported to be

$\frac FL=\mu _o\frac{I_1I_2}{2\pi d}$ .

But in Experiment 9 we have two circular coils. We may continue to make the approximation (implicit in the equation above) that the wire or, more relevantly, the coiled wire bundle has zero thickness. This introduces an error, to which one might return later for further consideration. (To proceed, presumably $d$ must be bigger than the actual thickness.) You could use Biot-Savart to figure out the force of circular coil #1 on coil #2. In the end, there would be an integral over coil #2 which would be trivial because the force is the same on each segment of coil #2. However, even if one is proficient at vectors and trigonometry, one is left with a one-dimensional integral (essentially over coil #1) that is not only beyond the scope of Phys.1 Anal.; it would even be a challenge for someone who has mastered ACM 95. While the answer cannot be represented in closed form with elementary functions, it can be reduced to a form that can be used to get the value with arbitrary numerical precision. The limiting behavior as $d/D\rightarrow 0$, where $D$ is the coil diameter, is just that given above, while the limiting behavior as $d/D\rightarrow \infty$ is the force between two magnetic dipoles or little bar magnets -- something that you could figure out. The actual answer for small $d/D$ is represented in the following graph.

This graph might prove useful in estimating one of the theoretical errors in your analysis of Experiment 9, and you could actually use the values from this graph to do a slightly more ambitious analysis than suggested by ZAP!

``...as a percent of the answer for the force."

The error associated with each quantity can be expressed as $\pm$ something with units the same as the quantity itself. However, it is far more convenient to express these as dimensionless, fractional or percentage uncertainties as follows:

$A\pm \Delta A=A(1\pm \frac{\Delta A}A)$ .

$\frac{\Delta A}A$ is the fractional error in $A$.

The next thing to do is to figure out the impact that the uncertainty in a certain quantity, call it $A$, has on your final answer, call it $F$. (In the case of Experiment 9, $F$ might be the magnetic force bewteen the two coils.) Assume for now that $A$ is the only source of error in $F$. One can substitute $A\pm \Delta A$ for $A$ in the formula for $F$, i.e., $F(A)$, and find the uncertainty in $F$:

$\Delta F=\vert F(A\pm \Delta A)-F(A)\vert$ .

To proceed, it is much simpler to assume something that is often the case, in particular, that all uncertainties are much smaller than the estimated average values. Then, only quantities up to first order in the errors (the $\Delta$'s) make sense. In that case, the procedure described above is just like taking a derivative:

$\Delta F\cong \vert\frac{dF}{dA}\vert\Delta A$

or, for the fractional error in $F$,

$\frac{\Delta F}F=\frac AF\frac{dF}{dA}\frac{\Delta A}A$ .

For example, if $F(A)=6A$, then $\frac{\Delta F}F=\frac{\Delta A}A$. However, if $F(A)=6A^5$, then $\frac{\Delta F}F=5\frac{\Delta A}A$, etc.

``If there is more than one source of error, then in a given measurement the errors may either add or cancel. Therefore the best error estimate for a number of errors is not the arithmetic sum of them. If the errors are particularly ``well-behaved'' in a statistical sense, it can be shown that the best estimate is the square root of the sum of the squares of the percent errors. Physicists very often assume that this method of error-combining is valid. For example, if you have estimated three uncertaqinties to be 10%, 7%, and 2%, the conventional estimate of the overall uncertainty is (153)$^{1/2}$%, or about 12%."

So for some function of two variables $F(A,B),$

$\frac{\Delta _{\text{total}}F}F=\sqrt{(\frac{\Delta _AF}F)^2+(\frac{\Delta _BF}F)^2}$

where the subscripts refer to the source of the fractional uncertainty in the final result.

``Notice that the small 2% uncertainty contributes very little."

Indeed, in the end, only the biggest one, or biggest ones really matter. The analysis helps identify just which those are. Of course, if you go back and improve the experiment, reducing those biggest errors will have the greatest impact on the quality of the result. However, after making such improvements, one might have to look more closely at sources of error and uncertainty that had previously been ``negligible.''

``What is your final estimate of the accuracy of the experiment?"