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What type of graph will it be What's the slope And error in the slope

Mathematics

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  1. Determine the type of graph.

    The experiment is for the "Verification of Ohm's Law," which states that voltage (VV) is directly proportional to current (II) for a given resistance (RR), i.e., V=IRV = IR. When plotting VV on the y-axis and II on the x-axis, this relationship results in a straight line.
    The graph will be a straight line.

  2. Calculate the slope of the V-I graph.

    The slope of a VV versus II graph represents the resistance RR. We will use linear regression to find the slope mm from the given data points (I,V)(I, V).
    The data points are:
    (Ii,Vi)(I_i, V_i)
    (0.12,0.00)(0.12, 0.00)
    (0.22,0.40)(0.22, 0.40)
    (0.32,0.60)(0.32, 0.60)
    (0.42,0.80)(0.42, 0.80)
    (0.52,1.00)(0.52, 1.00)
    (0.64,1.20)(0.64, 1.20)
    There are n=6n=6 data points.

    First, calculate the necessary sums:
    ∑Ii=0.12+0.22+0.32+0.42+0.52+0.64=2.24 A\sum I_i = 0.12 + 0.22 + 0.32 + 0.42 + 0.52 + 0.64 = 2.24 \, \text{A}
    ∑Vi=0.00+0.40+0.60+0.80+1.00+1.20=4.00 V\sum V_i = 0.00 + 0.40 + 0.60 + 0.80 + 1.00 + 1.20 = 4.00 \, \text{V}
    ∑(IiVi)=(0.12×0.00)+(0.22×0.40)+(0.32×0.60)+(0.42×0.80)+(0.52×1.00)+(0.64×1.20)\sum (I_i V_i) = (0.12 \times 0.00) + (0.22 \times 0.40) + (0.32 \times 0.60) + (0.42 \times 0.80) + (0.52 \times 1.00) + (0.64 \times 1.20)
    ∑(IiVi)=0.000+0.088+0.192+0.336+0.520+0.768=1.904 V⋅A\sum (I_i V_i) = 0.000 + 0.088 + 0.192 + 0.336 + 0.520 + 0.768 = 1.904 \, \text{V}\cdot\text{A}
    ∑(Ii2)=0.122+0.222+0.322+0.422+0.522+0.642\sum (I_i^2) = 0.12^2 + 0.22^2 + 0.32^2 + 0.42^2 + 0.52^2 + 0.64^2
    ∑(Ii2)=0.0144+0.0484+0.1024+0.1764+0.2704+0.4096=1.0216 A2\sum (I_i^2) = 0.0144 + 0.0484 + 0.1024 + 0.1764 + 0.2704 + 0.4096 = 1.0216 \, \text{A}^2

    The formula for the slope mm of a linear regression line is:

    m=n∑(IiVi)−∑Ii∑Vin∑(Ii2)−(∑Ii)2m = \frac{n \sum (I_i V_i) - \sum I_i \sum V_i}{n \sum (I_i^2) - (\sum I_i)^2}

    Substitute the calculated sums:

    m=6(1.904)−(2.24)(4.00)6(1.0216)−(2.24)2m = \frac{6(1.904) - (2.24)(4.00)}{6(1.0216) - (2.24)^2} m=11.424−8.966.1296−5.0176m = \frac{11.424 - 8.96}{6.1296 - 5.0176} m=2.4641.112m = \frac{2.464}{1.112} m≈2.215827 Ωm \approx 2.215827 \, \Omega

    Rounding to three decimal places:
    The slope is 2.216 Ω\boxed{\text{2.216 } \Omega}.

  3. Calculate the error in the slope (standard error of the slope).

    First, calculate the y-intercept cc:

    c=∑Vi−m∑Iinc = \frac{\sum V_i - m \sum I_i}{n} c=4.00−(2.215827)(2.24)6c = \frac{4.00 - (2.215827)(2.24)}{6} c=4.00−4.9634536=−0.9634536≈−0.160575 Vc = \frac{4.00 - 4.963453}{6} = \frac{-0.963453}{6} \approx -0.160575 \, \text{V}

    Next, calculate the sum of squared residuals, ∑(Vi−Vpi)2\sum (V_i - V_{pi})^2, where Vpi=mIi+cV_{pi} = m I_i + c:
    ∑(Vi−Vpi)2≈0.023362\sum (V_i - V_{pi})^2 \approx 0.023362 (calculated with full precision in thought process)
    The standard error of the estimate (ses_e) is:

    se=∑(Vi−Vpi)2n−2s_e = \sqrt{\frac{\sum (V_i - V_{pi})^2}{n-2}} se=0.0233626−2=0.0233624=0.0058405≈0.076423 Vs_e = \sqrt{\frac{0.023362}{6-2}} = \sqrt{\frac{0.023362}{4}} = \sqrt{0.0058405} \approx 0.076423 \, \text{V}

    The standard error of the slope (sms_m) is:

    sm=se∑(Ii2)−(∑Ii)2ns_m = \frac{s_e}{\sqrt{\sum (I_i^2) - \frac{(\sum I_i)^2}{n}}}

    The denominator term ∑(Ii2)−(∑Ii)2n\sum (I_i^2) - \frac{(\sum I_i)^2}{n} is:

    1.0216−(2.24)26=1.0216−5.01766=1.0216−0.8362666...≈0.18533331.0216 - \frac{(2.24)^2}{6} = 1.0216 - \frac{5.0176}{6} = 1.0216 - 0.8362666... \approx 0.1853333

    Now, calculate sms_m:

    sm=0.0764230.1853333s_m = \frac{0.076423}{\sqrt{0.1853333}} sm=0.0764230.4305035≈0.17752 Ωs_m = \frac{0.076423}{0.4305035} \approx 0.17752 \, \Omega

    Rounding to three decimal places:
    The error in the slope is 0.178 Ω\boxed{\text{0.178 } \Omega}.

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