Least Square Fit Workshop

PRINT 1st  STUDENT NAME:                                                    PRINT 2nd  STUDENT NAME:

The photoelectric effect experiment has you collect stopping potential V data for each color given as a frequency ν:

({{Color, yellow, green, violet, U . V .  1, U . V .  2}, {frequency ν \[Cross](10^14) Hz, 5.19, 5.49, 6.88, 7.41, 8.22}, {Stopping Voltage, 0.74, \[Placeholder], \[Placeholder], \[Placeholder], \[Placeholder]}})

According to the Einstein hypothesis

V = ((h )/e) ν - 
        W

a graph of stopping potential V versus frequency ν should be linear with a slope h/e.

Problem #1:  Assign variable names X and Y to the numerical values of  frequency ν and stopping potential V.  For example,

X[1] = 5.19*(10^14); and Y[1] = 
      0.74;

Include data from the other four points.

Problem #2:  Graph your data using

data = Table[{X[k], Y[k]}, {k, 1, 5}];
DataGraph = ListPlot[data]

The five data points should be (more or less)  on a straight line.

Problem #3:  Find the least square fit best straight line using

Num = 5;
 Δ  =  
        Num*(∑_(n = 1)^Num) 
              x[n]*x[n] - (∑_(k = 1)^Num) x[
                k]*(∑_(m = 1)^Num) x[m];

m = (1/Δ)(Num*  (∑_(n = 1)^Num) 
              y[n]*x[n] - (∑_(n = 1)^Num)  x[
                n] (∑_(n = 1)^Num) y[n])

b = (1/Δ) ((∑_(n = 1)^Num) 
              y[n]*  (∑_(n = 1)^Num) 
                  x[n]*x[n] - (∑_(n = 1)^Num) x[
                  n]* (∑_(n = 1)^Num) 
                  y[n]*x[n])

Problem #4:  Plot the least square fit best straight line using

TheoryGraph = Plot[m*x + b, {5.0*(10^14), 9.0*(10^14)}]

Plot the least square fit best straight line and the data together using

Show[TheoryGraph, DataGraph]

Problem #5:  Using the numerical value for the slope m=h/e and the value of the charge of an electron e, calculate Planck's constant h.

Problem #6:  Calculate the uncertainty in the slope δm using

δm = (Sqrt[(N/Δ)]) δV

where Δ is calculated in the Method of Least Squares, N=5 is the number of colors and δV is your estimate of the uncertainty in the voltage V.  Also calculate the uncertainty in Planck's constant δh using propagation of errors.  Report your measured value of Planck's constant as (Subscript[h, best]) ± δh.  Does the accepted value of Planck's constant (Subscript[h, accepted]) lie in this range?