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Maslowich
3 years ago
12

Two resistors are to be combined in parallelto form an equivalent resistance of 400Ω. The resistors are takenfrom available stoc

k on hand as acquired over the years. Readily available are two common resistorsrated at 500±50 Ωand two common resistors rated at 2000 Ω±5%. What isthe uncertainty in an equivalent 400 Ωresistance?(Hint: the equivalent resistance connected in parallel can be obtained by 1212TRRRRR=+)
Physics
1 answer:
vivado [14]3 years ago
6 0

Answer:

ΔR_{e} = 84   Ω,     R_{e} = (40 ± 8) 10¹   Ω

Explanation:

The formula for parallel equivalent resistance is

          1 / R_{e} = ∑ 1 / Ri

In our case we use a resistance of each

           R₁ = 500 ± 50  Ω

          R₂ = 2000 ± 5%

This percentage equals

        0.05 = ΔR₂ / R₂

        ΔR₂ = 0.05 R₂

        ΔR₂ = 0.05 2000 = 100   Ω

We write the resistance

        R₂ = 2000 ± 100    Ω

We apply the initial formula

        1 / R_{e} = 1 / R₁ + 1 / R₂

        1 / R_{e} = 1/500 + 1/2000 = 0.0025

        R_{e}  = 400    Ω

Let's look for the error  (uncertainly) of Re

      R_{e} = R₁R₂ / (R₁ + R₂)

       R’= R₁ + R₂

       R_{e} = R₁R₂ / R’

Let's look for the uncertainty of this equation

      ΔR_{e} / R_{e} = ΔR₁ / R₁ + ΔR₂ / R₂ + ΔR’/ R’

The uncertainty of a sum is

      ΔR’= ΔR₁ + ΔR₂

We substitute the values

     ΔR_{e} / 400 = 50/500 + 100/2000 + (50 +100) / (500 + 2000)

     ΔR_{e} / 400 = 0.1 + 0.05 + 0.06

     ΔR_{e} = 0.21 400

     ΔR_{e} = 84   Ω

Let's write the resistance value with the correct significant figures

    R_{e} = (40 ± 8) 10¹   Ω

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