PHYS 211 Final, Fall 2005

1 Physics 211L Final Exam January 2006 Name:________________________________ID__________________ Section: M-1 M-3 T-1 T-3 W-1 W-3 Th-1 Th-3 F-1 F-3 1. The magnetic field due to a long straight current -carrying wire is measured by means of a Tesla meter, which has not been properly zeroed (Ignore the magnetic field of the Earth). The magnetic field H and the probe-head distance from the wire d are recorded, with the following results: H (mT) d(mm) 3.42 2.0 2.26 3.1 1.80 3.9 1.55 5.1 1.32 6.2 The current is kept constant at 30.1 A; the ammeter tolerance is 2%. Write the relation between H and d. Choose the proper variables and use linear regression to find the permeability of free space ( 0 ) and the meter reading when it is very far from the wire. Compare 0 to the literature value x 10 -7 SI. (60%) 2 3 2. How did you measure currents in the PHYS211L DMM experiment? (10%) 3. Why is the electron track visible in your PHYS211L e/m experiment? The electrons circulate counterclockwise, looking from above; what is the direction of the magnetic field? (10%) 4 4. In your best PHYS211L transformer configuration the primary coil has 400 turns while the secondary has 1600 turns. The input voltage is 1 Vp-p and the input current is 20 mA at 70Hz. What are the expected values of the open circuit voltage and short circuit current in the secondary coil? Do you expect the measured values to be more or less than expected? Why? (12%) 5. Why can you not use the PHYS 211LWheatstone Bridge to measure the resistances of your metal rods? (8 %) 5 LINEAR REGRESSION Many experiments yield a series of pairs of data values. Usually the xi values are selected and the yi values are measured. The method of least squares is used to fit a curve (find a theoretical equation) to a set of experimental data. First assume that a linear relation exists between y and x y = Ax + B (1) Substitution of x = xi will in general not give the value of yi ei = y yi = Axi + B yi (2) To determine the best straight line that fits the N, sets of data, A and B have to be chosen simultaneous equations, obtained by equating the partial derivatives of (y yi)2 with respect to A and B to zero, should be solved. This condition leads then to the following results ∆ − = ∑ ∑ ∑ i i i i y x x y N A ) ( (3) and ∆ − = ∑ ∑ ∑ ∑ ) ( i i i i i x y x y x B 2 (4) where 2 2 (∑ ) ∑ − ∆ = i i x x N (5) The correlation coefficient r provides an indicator of how good a fit the best straight line is. This coefficient is defined as ∑ ∑ ∑ − − − − = 2 2 ) ( ) ( ) )( ( y y x x y y x x r i i i i (6) For r = 0, the values of x and y are independent of one another and there is no linear correlation. The closer r is to +1 or to 1, the better the linear correlation is. Finally, the errors in A and B are given by: ∆ − = ∑ 2 2 2 i A e N N σ N xi A B 2 2 2 = σ Σ σ (7)