This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Step 1: Determine the type of graph. The experiment is for the "Verification of Ohm's Law," which states that voltage (V) is directly proportional to current (I) for a given resistance (R), i.e., V = IR. When plotting V on the y-axis and I on the x-axis, this relationship results in a straight line. The graph will be a straight line. Step 2: Calculate the slope of the V-I graph. The slope of a V versus I graph represents the resistance R. We will use linear regression to find the slope m from the given data points (I, V). The data points are: (I_i, V_i) (0.12, 0.00) (0.22, 0.40) (0.32, 0.60) (0.42, 0.80) (0.52, 1.00) (0.64, 1.20) There are n=6 data points. First, calculate the necessary sums: I_i = 0.12 + 0.22 + 0.32 + 0.42 + 0.52 + 0.64 = 2.24 \, A V_i = 0.00 + 0.40 + 0.60 + 0.80 + 1.00 + 1.20 = 4.00 \, V (I_i V_i) = (0.12 × 0.00) + (0.22 × 0.40) + (0.32 × 0.60) + (0.42 × 0.80) + (0.52 × 1.00) + (0.64 × 1.20) (I_i V_i) = 0.000 + 0.088 + 0.192 + 0.336 + 0.520 + 0.768 = 1.904 \, V·A (I_i^2) = 0.12^2 + 0.22^2 + 0.32^2 + 0.42^2 + 0.52^2 + 0.64^2 (I_i^2) = 0.0144 + 0.0484 + 0.1024 + 0.1764 + 0.2704 + 0.4096 = 1.0216 \, A^2 The formula for the slope m of a linear regression line is: m = (n (I_i V_i) - I_i V_i)/(n (I_i^2) - ( I_i)^2) Substitute the calculated sums: m = (6(1.904) - (2.24)(4.00))/(6(1.0216) - (2.24)^2) m = (11.424 - 8.96)/(6.1296 - 5.0176) m = (2.464)/(1.112) m ≈ 2.215827 \, Rounding to three decimal places: The slope is 2.216 . Step 3: Calculate the error in the slope (standard error of the slope). First, calculate the y-intercept c: c = ( V_i - m I_i)/(n) c = (4.00 - (2.215827)(2.24))/(6) c = (4.00 - 4.963453)/(6) = (-0.963453)/(6) ≈ -0.160575 \, V Next, calculate the sum of squared residuals, (V_i - V_pi)^2, where V_pi = m I_i + c: (V_i - V_pi)^2 ≈ 0.023362 (calculated with full precision in thought process) The standard error of the estimate (s_e) is: s_e = sqrt( (V_i - V_pi))^2n-2 s_e = sqrt((0.023362)/(6-2)) = sqrt((0.023362)/(4)) = sqrt(0.0058405) ≈ 0.076423 \, V The standard error of the slope (s_m) is: s_m = (s_e)/(sqrt( (I_i^2) - ( I_i)^2)n) The denominator term (I_i^2) - (( I_i)^2)/(n) is: 1.0216 - ((2.24)^2)/(6) = 1.0216 - (5.0176)/(6) = 1.0216 - 0.8362666... ≈ 0.1853333 Now, calculate s_m: s_m = (0.076423)/(sqrt(0.1853333)) s_m = (0.076423)/(0.4305035) ≈ 0.17752 \, Rounding to three decimal places: The error in the slope is 0.178 . Send me the next one 📸