PHYS272_Lab5

=Physics for Scientists and Engineers – Lab 5: PreLab =  Section 6-7 of the book Experimentation: An Introduction to Measurement Theory and Experiment Design shows how to do a least squares fit of data to a straight line and gives equations for finding the slope, y-intercept and the errors in these values. These equations assume that the error in the y values (S y ) for each data point is the same, which is often not the case. If the errors are different for each data point, we should use a //weighted// least squares fit, described on pg 184-185 of Experimentation.  Let's try an example. Suppose you have the following x and y data So when x = 1.00, y = 3.89 ± 0.05  1. Fill in the column of weights in the table above. Each weight is given by math w_i = 1/(S_{y_i})^2 math
 * ~ x ||~ y ||~ S y ||~ w ||
 * 1.00 ||> 3.89 || ± 0.05 ||  ||
 * 2.00 ||> 6.19 || ± 0.08 ||  ||
 * 3.00 ||> 8.65 || ± 0.10 ||  ||
 * 4.00 || 10.97 || ± 0.12 ||  ||
 * 5.00 || 13.16 || ± 0.04 ||  ||

2. Find the weighted slope and y-intercept of the data. These are given by math \displaystyle m = \frac{\sum w_i \sum w_i x_i y_i - \sum w_i x_i \sum w_i y_i}{\sum w_i \sum w_i x_i^2 - (\sum w_i x_i)^2} math

math \displaystyle b = \frac{\sum w_i y_i \sum w_i x_i^2 - \sum w_i x_i \sum w_i x_i y_i}{\sum w_i \sum w_i x_i^2 - (\sum w_i x_i)^2} math

3. Calculate the deviations of the data points from your straight line fit δ i = y i - (m x i + b) and the standard deviation of those deviations math \displaystyle S_y = \sqrt{\frac{\sum w_i \delta_i^2}{N-2}} math where N is the number of data points.

4. Finally, calculate the uncertainty in the slope and y-intercept (S m and S b ) via math \displaystyle \bar{x} = \frac{\sum w_i x_i}{\sum w_i} math

math W = \sum(w_i(x_i - \bar{x})^2) math

math S_m^2 = S_y^2/W math

math \displaystyle S_b^2 = S_y^2 \left(\frac{1}{\sum w_i} + \frac{{\bar{x}}^2}{W} \right) math You may find these expressions easier to evaluate using a spreadsheet or similar. =Physics for Scientists and Engineers – Lab 5 =

Objective:

 * To measure the motional emf induced on a coil passing through a magnetic field.
 * To measure the value of e/m by measuring the radius of curvature of an electron beam

Equipment:

 * Motion sensor
 * Pasco 1.2 m dynamics track
 * Dynamics cart with coil mounting bracket and rectangular coil
 * Permanent magnet
 * Banana clips
 * Bell Gauss meters with Hall element probes
 * Science Workshop and Graphical Analysis software
 * e/m apparatus

**Motional Electromotive Force **
 When a conducting wire, attached to a circuit, is passed through a perpendicular magnetic field, the induced emf (voltage) on the wire is given by,

math V=Blv \quad \quad \mathrm{(eq 1)} math

where B is the magnetic field (in Tesla), l is the length of the wire, and v is the speed of the wire through the magnetic field. For a coil, with N turns, the induced emf is,

math V=NBlv \quad \quad \mathrm{(eq 2)} math

If the maximum induced voltage is measured for different velocities of the coil through the magnet, a plot of maximum voltage (y-axis) vs. velocity (x-axis) will yield a slope equal to N B l. Thus, the average magnetic field of the magnet can be determined by dividing the slope of the graph by N l.

**Circular Motion in a Magnetic Field **
 When an electron moves perpendicular to a magnetic field B it will travel in a circle of radius r determined by

math \displaystyle qvB=\frac{mv^2}{r} \quad \quad \mathrm{(eq 3)} math

An electron accelerated from rest through a potential difference of V will emerge with a speed given by

math \displaystyle \frac{1}{2}mv^2=qV \quad \quad \mathrm{(eq 4)} math

From these two results

math q^2B^2r^2=m^2v^2=m2qV \quad \quad \mathrm{(eq 5)} math

so, when q=e

math \displaystyle \frac{e}{m}=\frac{2V}{B^2r^2} \quad \quad \mathrm{(eq 6)} math

An extremely uniform magnetic field is generated by a pair of coils, the Helmholtz coils, where the distance between the coils equals the radius R of the coils. The magnetic field at the origin for a coil of N turns that are parallel to the x-y plane and are at a distance z from the origin is

math \displaystyle B=\frac{\mu_0 N I R^2}{2(z^2+R^2)^{3/2}} \quad\quad \mathrm{(eq 7)} math

A similar result gives the magnetic field of a second coil at -z, adding the fields of the two coils gives

math \displaystyle B=\frac{\mu_0 N I R^2}{(z^2+R^2)^{3/2}} \quad \quad \mathrm{(eq 8)} math

As a good calculus example it can be shown that for small distances from the origin along the z axis the first three derivatives in a Taylor series expansion are zero when the distances from the origin of the coils are |z| = R/2. Thus coils at this separation produce nearly uniform fields between the coils. Inserting this into equation (7) gives a magnetic field at the center that is

math \displaystyle B=\left(\frac{4}{5} \right)^{3/2} \frac{\mu_0 N I}{R} \quad \quad \mathrm{(eq 9)} math

Electromotive Force
 Mount the rectangular coil onto the dynamics cart and attach the ends of the wires of the coil to the Science Workshop interface with the banana clips and wires provided. The far edge of the coil should be centered vertically between the poles of the permanent magnet. The number of turns, N, of the coil is 100. Measure the outer and inner widths of the coil (of the narrow side), calculate the average width, l and record these values in your lab book.  Using a Bell Hall element gaussmeter, measure and record the magnetic field at the center of the magnet. Use the same procedure that you used in lab 4, take 5 measurements and calculate the mean and the standard deviation of the mean to use as the uncertainty. You will compare this measured value of the magnetic field with the experimentally determined value from Equation (2). <span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> Open Pasco Capstone and plug the motion sensor into digital channels 1 and 2, and the voltage sensor into analogue channel A. Set the motion sensor for 50 measurements per second, and the voltage sensor at 1000 measurements per second Under sampling options, set stop time to 3 seconds. Open a graph of voltage versus time, and add velocity versus time. Select mean and standard deviation statistics for each. Click on the record icon and tap the launching post on the dynamics cart to launch the cart toward the motion sensor. //Make sure to stop the cart before it runs off the end of the track or hits the motion sensor.// <span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> On the voltage vs. time graph, click and drag the cursor to select a portion of the top section of the graph that is approximately constant for the maximum voltage. Read the mean and standard deviation for V max from the statistics. Find the percent error of the measurement by taking the standard deviation divided by the mean and multiplied by 100%. On the velocity vs. time graph, measure the velocity of the cart at the point of maximum voltage. Calculate and record the value of V max / v, and compare with the measured value N B l, giving the percent difference. <span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> Repeat the previous measurement and analysis for four more data runs, varying the speed of the cart for each run. <span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> Open Graphical Analysis and make a graph of voltage vs. velocity. Include error bars on your plot, using the standard deviations of the voltage and velocity measurements as the uncertainties. Fit the data to a straight line and record the value of the slope. Using the value of the slope and Equation (2), determine the magnetic field B. Compare the determined value with the value measured by the gaussmeter, and compute the percentage error. Also calculate the weighted slope using the equation from the prelab. How does this compare with the unweighted slope? <span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;">
 * Question:** How does the percentage error of the magnetic field determined from the slope of the graph compare to the percentage error calculated from the individual data points? Explain.

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">e/m Experiment
<span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> Set up the e/m apparatus as shown on the flier in the lab. Turn on the filament heater and electron acceleration voltage. Record the voltages of the heater and accelerator. You should see a fine purple line where the electrons pass through the tube. **Caution: Do not leave the electron beam on in this geometry for very long, as the electrons will eventually destroy the tube where they hit.** Take a small magnet and move it close to the tube. Describe in your lab report what you see. <span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> Turn on the voltage to the Helmholtz coils. Record the current in the Helmholtz coils. Calculate the magnetic field generated. Measure the radius of curvature of the electron beam by taking the average of the positions where the electron beam crosses the ruler on the right and the left. <span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> Repeat for four different magnetic field values. Generate a linear plot and use the slope of that plot to determine the value of e/m for an electron, and compare with the accepted value of e/m.