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Calculate the resistance and its uncertainty using Ohm's Law, given V = (18 ± 3) V and I = (5 ± 0.1) A.

Physics

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Step-by-step solution

5 steps
  1. Identify the given values and their absolute errors.

    The potential difference is V=(18±3) VV = (18 \pm 3) \text{ V}.
    So, the measured potential difference is V=18 VV = 18 \text{ V} and its absolute error is ΔV=3 V\Delta V = 3 \text{ V}.

    The current is I=(5±0.1) AI = (5 \pm 0.1) \text{ A}.
    So, the measured current is I=5 AI = 5 \text{ A} and its absolute error is ΔI=0.1 A\Delta I = 0.1 \text{ A}.

  2. Recall the formula for resistance and the rule for propagation of errors in division.

    According to Ohm's Law, resistance RR is given by:

    R=VIR = \frac{V}{I}

    When quantities are divided, the maximum fractional error in the result is the sum of the fractional errors of the individual quantities.

    ΔRR=ΔVV+ΔII\frac{\Delta R}{R} = \frac{\Delta V}{V} + \frac{\Delta I}{I}
  3. Calculate the fractional errors for potential difference and current.

    Fractional error in potential difference:

    ΔVV=3 V18 V=16\frac{\Delta V}{V} = \frac{3 \text{ V}}{18 \text{ V}} = \frac{1}{6}

    Fractional error in current:

    ΔII=0.1 A5 A=150\frac{\Delta I}{I} = \frac{0.1 \text{ A}}{5 \text{ A}} = \frac{1}{50}
  4. Calculate the total fractional error in resistance.

    ΔRR=16+150\frac{\Delta R}{R} = \frac{1}{6} + \frac{1}{50}

    To add these fractions, find a common denominator, which is 150:

    ΔRR=25150+3150=28150=1475\frac{\Delta R}{R} = \frac{25}{150} + \frac{3}{150} = \frac{28}{150} = \frac{14}{75}
  5. Convert the total fractional error to percentage error.

    Percentage error is the fractional error multiplied by 100%100\%.

    Percentage error=ΔRR×100%\text{Percentage error} = \frac{\Delta R}{R} \times 100\% Percentage error=1475×100%\text{Percentage error} = \frac{14}{75} \times 100\% Percentage error=14×43%=563%\text{Percentage error} = \frac{14 \times 4}{3}\% = \frac{56}{3}\% Percentage error≈18.666...%\text{Percentage error} \approx 18.666...\%

    Rounding to one decimal place, the percentage error is 18.7%18.7\%.

    The percentage error in the measurement of resistance of the conductor is 18.7%\boxed{\text{18.7\%}}.

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