PHYS 205L Final, Fall 2007

Physics 205L Final exam January 17, 2008 Name:____________________________ Section number:___________________ Instructor’s Name:__________________ ID number:_________________________ DO NOT START THE EXAM BEFORE YOU ARE TOLD TO BEGIN Grading I II-1 II-2 II-3 TOTAL The duration of this exam is 60 minutes. No notes or books allowed. Scientific calculators are allowed All results should be given with the exact number of significant figures. 1 I. (55%) The dielectric constant k of the material that fills the space between the two conductors of a coaxial cable is to be determined. Cables of different outer diameters b were used and the capacitance of each was measured. All cables have an inner diameter of a = 0.037 cm and a length L = 1 +/- 0.001 m. The permittivity of free space is ε0= 2 2 8 85 10 12 C / Nm . − × The results below were obtained: b [cm] C(pF) 0.110 102.10 0.231 60.72 0.320 51.55 0.415 46.01 0.517 42.17 You may find the formulae on page 5 useful. a- Write down the necessary relation between C and b. b- Choose your variables x and y such that you obtain a linear relationship between them. c- Use linear regression to find the slope of the line along with its error. 2 d- Determine the dielectric constant k along with its error. 3 4 Linear Regression 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. The “errors” will be 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 so that the sum of the squares of the “errors” is minimized. This means that the 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 Δ − = ∑ ∑ ∑ i i i i y x ( x y ) N A (3) and Δ − = ∑ ∑ ∑ ∑ ) ( i i i i i x y x y x B 2 (4) where 2 2 (∑ ) ∑ − Δ = 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 error in A is given by: Δ − = ∑ 2 2 2 i A e N N σ 5 II. Questions 1. In the “Planck’s Constant Experiment” (15%) a- We used the equation e e f h VS − φ = ( ) in order to determine h andφ . Explain how.(10%) b- Define the photoelectric effect. (5%) 2. In the e/m experiment, why did we tilt the Helmholtz coils and kept them at that orientation throughout the experiment? (10%) 6 7 3. In “Ohm’s Law” experiment, explain how the resistivity of metal rods were determined. Include necessary equations. (20%)